Theory "real_topology"

Signature

Definitions

within

⊢ ∀net s. net within s = mk_net (λx y. netord net x y ∧ x ∈ s)

uniformly_continuous_on

⊢ ∀f s.
      f uniformly_continuous_on s ⇔
      ∀e.
          0 < e ⇒
          ∃d.
              0 < d ∧
              ∀x x'. x ∈ s ∧ x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e

trivial_limit

⊢ ∀net.
      trivial_limit net ⇔
      (∀a b. a = b) ∨ ∃a b. a ≠ b ∧ ∀x. ¬netord net x a ∧ ¬netord net x b

tendsto

⊢ ∀f l net. (f --> l) net ⇔ ∀e. 0 < e ⇒ eventually (λx. dist (f x,l) < e) net

sums

⊢ ∀f l s. (f sums l) s ⇔ ((λn. sum (s ∩ (0 .. n)) f) --> l) sequentially

summable

⊢ ∀s f. summable s f ⇔ ∃l. (f sums l) s

subtopology

⊢ ∀top u. subtopology top u = topology {s ∩ u | open_in top s}

subspace

⊢ ∀s.
      subspace s ⇔
      0 ∈ s ∧ (∀x y. x ∈ s ∧ y ∈ s ⇒ x + y ∈ s) ∧ ∀c x. x ∈ s ⇒ c * x ∈ s

sphere

⊢ ∀x e. sphere (x,e) = {y | dist (x,y) = e}

span

⊢ ∀s. span s = subspace hull s

setdist

⊢ ∀s t.
      setdist (s,t) =
      if (s = ∅) ∨ (t = ∅) then 0 else inf {dist (x,y) | x ∈ s ∧ y ∈ t}

sequentially

⊢ sequentially = mk_net (λm n. m ≥ n)

permutes

⊢ ∀p s. p permutes s ⇔ (∀x. x ∉ s ⇒ (p x = x)) ∧ ∀y. ∃!x. p x = y

pairwise

⊢ ∀r s. pairwise r s ⇔ ∀x y. x ∈ s ∧ y ∈ s ∧ x ≠ y ⇒ r x y

open_segment

⊢ ∀a b. segment (a,b) = segment [(a,b)] DIFF {a; b}

OPEN_interval

⊢ ∀a b. interval (a,b) = {x | a < x ∧ x < b}

open_def

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. dist (x',x) < e ⇒ x' ∈ s

netlimit

⊢ ∀net. netlimit net = @a. ∀x. ¬netord net x a

net_TY_DEF

⊢ ∃rep. TYPE_DEFINITION isnet rep

midpoint

⊢ ∀a b. midpoint (a,b) = 2⁻¹ * (a + b)

locally

⊢ ∀P s.
      locally P s ⇔
      ∀w x.
          open_in (subtopology euclidean s) w ∧ x ∈ w ⇒
          ∃u v.
              open_in (subtopology euclidean s) u ∧ P v ∧ x ∈ u ∧ u ⊆ v ∧
              v ⊆ w

linear

⊢ ∀f. linear f ⇔ (∀x y. f (x + y) = f x + f y) ∧ ∀c x. f (c * x) = c * f x

limit_point_of

⊢ ∀x s. x limit_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ ∃y. y ≠ x ∧ y ∈ s ∧ y ∈ t

lim_def

⊢ ∀net f. lim net f = @l. (f --> l) net

isnet

⊢ ∀g. isnet g ⇔ ∀x y. (∀z. g z x ⇒ g z y) ∨ ∀z. g z y ⇒ g z x

is_interval

⊢ ∀s.
      is_interval s ⇔
      ∀a b x. a ∈ s ∧ b ∈ s ⇒ a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a ⇒ x ∈ s

interior

⊢ ∀s. interior s = {x | ∃t. open t ∧ x ∈ t ∧ t ⊆ s}

infsum

⊢ ∀s f. infsum s f = @l. (f sums l) s

independent

⊢ ∀s. independent s ⇔ ¬dependent s

in_direction

⊢ ∀a v. a in_direction v = at a within {b | ∃c. 0 ≤ c ∧ (b − a = c * v)}

homeomorphism

⊢ ∀s t f g.
      homeomorphism (s,t) (f,g) ⇔
      (∀x. x ∈ s ⇒ (g (f x) = x)) ∧ (IMAGE f s = t) ∧ f continuous_on s ∧
      (∀y. y ∈ t ⇒ (f (g y) = y)) ∧ (IMAGE g t = s) ∧ g continuous_on t

homeomorphic

⊢ ∀s t. s homeomorphic t ⇔ ∃f g. homeomorphism (s,t) (f,g)

hausdist

⊢ ∀s t.
      hausdist (s,t) =
      if
        {setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t} ≠ ∅ ∧
        ∃b.
            ∀d.
                d ∈ {setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t} ⇒
                d ≤ b
      then
        sup ({setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t})
      else 0

gdelta

⊢ ∀s. gdelta s ⇔ ∃g. COUNTABLE g ∧ (∀u. u ∈ g ⇒ open u) ∧ (BIGINTER g = s)

fsigma

⊢ ∀s. fsigma s ⇔ ∃g. COUNTABLE g ∧ (∀c. c ∈ g ⇒ closed c) ∧ (BIGUNION g = s)

frontier

⊢ ∀s. frontier s = closure s DIFF interior s

eventually

⊢ ∀p net.
      eventually p net ⇔
      trivial_limit net ∨ ∃y. (∃x. netord net x y) ∧ ∀x. netord net x y ⇒ p x

euclidean

⊢ euclidean = topology open

dist

⊢ ∀x y. dist (x,y) = abs (x − y)

dim

⊢ ∀v. dim v = @n. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b ∧ b HAS_SIZE n

diameter

⊢ ∀s. diameter s = if s = ∅ then 0 else sup {abs (x − y) | x ∈ s ∧ y ∈ s}

dependent

⊢ ∀s. dependent s ⇔ ∃a. a ∈ s ∧ a ∈ span (s DELETE a)

continuous_on

⊢ ∀f s.
      f continuous_on s ⇔
      ∀x.
          x ∈ s ⇒
          ∀e.
              0 < e ⇒
              ∃d. 0 < d ∧ ∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e

continuous

⊢ ∀f net. f continuous net ⇔ (f --> f (netlimit net)) net

connected_component

⊢ ∀s x y. connected_component s x y ⇔ ∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t

connected

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          open e1 ∧ open e2 ∧ s ⊆ e1 ∪ e2 ∧ (e1 ∩ e2 ∩ s = ∅) ∧ e1 ∩ s ≠ ∅ ∧
          e2 ∩ s ≠ ∅

condensation_point_of

⊢ ∀x s. x condensation_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ ¬COUNTABLE (s ∩ t)

components

⊢ ∀s. components s = {connected_component s x | x | x ∈ s}

complete

⊢ ∀s.
      complete s ⇔
      ∀f. (∀n. f n ∈ s) ∧ cauchy f ⇒ ∃l. l ∈ s ∧ (f --> l) sequentially

compact

⊢ ∀s.
      compact s ⇔
      ∀f.
          (∀n. f n ∈ s) ⇒
          ∃l r. l ∈ s ∧ (∀m n. m < n ⇒ r m < r n) ∧ (f ∘ r --> l) sequentially

collinear

⊢ ∀s. collinear s ⇔ ∃u. ∀x y. x ∈ s ∧ y ∈ s ⇒ ∃c. x − y = c * u

closure

⊢ ∀s. closure s = s ∪ {x | x limit_point_of s}

closest_point

⊢ ∀s a. closest_point s a = @x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,x) ≤ dist (a,y)

closed_segment

⊢ ∀l. segment l = {(1 − u) * FST (HD l) + u * SND (HD l) | 0 ≤ u ∧ u ≤ 1}

CLOSED_interval

⊢ ∀l. interval l = {x | FST (HD l) ≤ x ∧ x ≤ SND (HD l)}

closed_def

⊢ ∀s. closed s ⇔ open (𝕌(:real) DIFF s)

cball

⊢ ∀x e. cball (x,e) = {y | dist (x,y) ≤ e}

cauchy

⊢ ∀s. cauchy s ⇔ ∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ∧ n ≥ N ⇒ dist (s m,s n) < e

bounded_def

⊢ ∀s. bounded s ⇔ ∃a. ∀x. x ∈ s ⇒ abs x ≤ a

bilinear

⊢ ∀f. bilinear f ⇔ (∀x. linear (λy. f x y)) ∧ ∀y. linear (λx. f x y)

between

⊢ ∀x a b. between x (a,b) ⇔ (dist (a,b) = dist (a,x) + dist (x,b))

ball

⊢ ∀x e. ball (x,e) = {y | dist (x,y) < e}

at_posinfinity

⊢ at_posinfinity = mk_net (λx y. x ≥ y)

at_neginfinity

⊢ at_neginfinity = mk_net (λx y. x ≤ y)

at_infinity

⊢ at_infinity = mk_net (λx y. abs x ≥ abs y)

at

⊢ ∀a. at a = mk_net (λx y. 0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a))

Theorems

WLOG_LT

⊢ (∀m. P m m) ∧ (∀m n. P m n ⇔ P n m) ∧ (∀m n. m < n ⇒ P m n) ⇒ ∀m y. P m y

WLOG_LE

⊢ (∀m n. P m n ⇔ P n m) ∧ (∀m n. m ≤ n ⇒ P m n) ⇒ ∀m n. P m n

WITHIN_WITHIN

⊢ ∀net s t. (net within s) within t = net within s ∩ t

WITHIN_UNIV

⊢ ∀x. at x within 𝕌(:real) = at x

WITHIN

⊢ ∀n s x y. netord (n within s) x y ⇔ netord n x y ∧ x ∈ s

URYSOHN_STRONG

⊢ ∀s t a b.
      closed s ∧ closed t ∧ (s ∩ t = ∅) ∧ a ≠ b ⇒
      ∃f.
          f continuous_on 𝕌(:real) ∧ (∀x. f x ∈ segment [(a,b)]) ∧
          (∀x. (f x = a) ⇔ x ∈ s) ∧ ∀x. (f x = b) ⇔ x ∈ t

URYSOHN_LOCAL_STRONG

⊢ ∀s t u a b.
      closed_in (subtopology euclidean u) s ∧
      closed_in (subtopology euclidean u) t ∧ (s ∩ t = ∅) ∧ a ≠ b ⇒
      ∃f.
          f continuous_on u ∧ (∀x. x ∈ u ⇒ f x ∈ segment [(a,b)]) ∧
          (∀x. x ∈ u ⇒ ((f x = a) ⇔ x ∈ s)) ∧ ∀x. x ∈ u ⇒ ((f x = b) ⇔ x ∈ t)

URYSOHN_LOCAL

⊢ ∀s t u a b.
      closed_in (subtopology euclidean u) s ∧
      closed_in (subtopology euclidean u) t ∧ (s ∩ t = ∅) ⇒
      ∃f.
          f continuous_on u ∧ (∀x. x ∈ u ⇒ f x ∈ segment [(a,b)]) ∧
          (∀x. x ∈ s ⇒ (f x = a)) ∧ ∀x. x ∈ t ⇒ (f x = b)

URYSOHN

⊢ ∀s t a b.
      closed s ∧ closed t ∧ (s ∩ t = ∅) ⇒
      ∃f.
          f continuous_on 𝕌(:real) ∧ (∀x. f x ∈ segment [(a,b)]) ∧
          (∀x. x ∈ s ⇒ (f x = a)) ∧ ∀x. x ∈ t ⇒ (f x = b)

UPPER_LOWER_HEMICONTINUOUS_EXPLICIT

⊢ ∀f t s.
      (∀x. x ∈ s ⇒ f x ⊆ t) ∧
      (∀u.
           open_in (subtopology euclidean t) u ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ∧
      (∀u.
           closed_in (subtopology euclidean t) u ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇒
      ∀x e.
          x ∈ s ∧ 0 < e ∧ bounded (f x) ∧ f x ≠ ∅ ⇒
          ∃d.
              0 < d ∧
              ∀x'.
                  x' ∈ s ∧ dist (x,x') < d ⇒
                  (∀y. y ∈ f x ⇒ ∃y'. y' ∈ f x' ∧ dist (y,y') < e) ∧
                  ∀y'. y' ∈ f x' ⇒ ∃y. y ∈ f x ∧ dist (y',y) < e

UPPER_LOWER_HEMICONTINUOUS

⊢ ∀f t s.
      (∀x. x ∈ s ⇒ f x ⊆ t) ∧
      (∀u.
           open_in (subtopology euclidean t) u ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ∧
      (∀u.
           closed_in (subtopology euclidean t) u ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇒
      ∀x e.
          x ∈ s ∧ 0 < e ∧ bounded (f x) ⇒
          ∃d. 0 < d ∧ ∀x'. x' ∈ s ∧ dist (x,x') < d ⇒ hausdist (f x,f x') < e

UPPER_HEMICONTINUOUS

⊢ ∀f t s.
      (∀x. x ∈ s ⇒ f x ⊆ t) ⇒
      ((∀u.
            open_in (subtopology euclidean t) u ⇒
            open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇔
       ∀u.
           closed_in (subtopology euclidean t) u ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∩ u ≠ ∅})

UPPER_BOUND_FINITE_SET_REAL

⊢ ∀f s. FINITE s ⇒ ∃a. ∀x. x ∈ s ⇒ f x ≤ a

UPPER_BOUND_FINITE_SET

⊢ ∀f s. FINITE s ⇒ ∃a. ∀x. x ∈ s ⇒ f x ≤ a

UNIT_INTERVAL_NONEMPTY

⊢ interval [(0,1)] ≠ ∅ ∧ interval (0,1) ≠ ∅

UNION_INTERIOR_SUBSET

⊢ ∀s t. interior s ∪ interior t ⊆ interior (s ∪ t)

UNION_FRONTIER

⊢ ∀s t.
      frontier s ∪ frontier t =
      frontier (s ∪ t) ∪ frontier (s ∩ t) ∪ frontier s ∩ frontier t

UNIFORMLY_CONVERGENT_EQ_CAUCHY_ALT

⊢ ∀P s.
      (∃l. ∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e) ⇔
      ∀e.
          0 < e ⇒
          ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ m < n ∧ P x ⇒ dist (s m x,s n x) < e

UNIFORMLY_CONVERGENT_EQ_CAUCHY

⊢ ∀P s.
      (∃l. ∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e) ⇔
      ∀e. 0 < e ⇒ ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x ⇒ dist (s m x,s n x) < e

UNIFORMLY_CONTINUOUS_ON_VMUL

⊢ ∀s c v.
      c uniformly_continuous_on s ⇒ (λx. c x * v) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_SUM

⊢ ∀t f s.
      FINITE s ∧ (∀a. a ∈ s ⇒ f a uniformly_continuous_on t) ⇒
      (λx. sum s (λa. f a x)) uniformly_continuous_on t

UNIFORMLY_CONTINUOUS_ON_SUBSET

⊢ ∀f s t. f uniformly_continuous_on s ∧ t ⊆ s ⇒ f uniformly_continuous_on t

UNIFORMLY_CONTINUOUS_ON_SUB

⊢ ∀f g s.
      f uniformly_continuous_on s ∧ g uniformly_continuous_on s ⇒
      (λx. f x − g x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_SETDIST

⊢ ∀s t. (λy. setdist ({y},s)) uniformly_continuous_on t

UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY

⊢ ∀f s.
      f uniformly_continuous_on s ⇔
      ∀x y.
          (∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧ ((λn. x n − y n) --> 0) sequentially ⇒
          ((λn. f (x n) − f (y n)) --> 0) sequentially

UNIFORMLY_CONTINUOUS_ON_NEG

⊢ ∀f s. f uniformly_continuous_on s ⇒ (λx. -f x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_MUL

⊢ ∀f g s.
      f uniformly_continuous_on s ∧ g uniformly_continuous_on s ∧
      bounded (IMAGE f s) ∧ bounded (IMAGE g s) ⇒
      (λx. f x * g x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_ID

⊢ ∀s. (λx. x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_EQ

⊢ ∀f g s.
      (∀x. x ∈ s ⇒ (f x = g x)) ∧ f uniformly_continuous_on s ⇒
      g uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_DIST_CLOSEST_POINT

⊢ ∀s t.
      closed s ∧ s ≠ ∅ ⇒
      (λx. dist (x,closest_point s x)) uniformly_continuous_on t

UNIFORMLY_CONTINUOUS_ON_CONST

⊢ ∀s c. (λx. c) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_COMPOSE

⊢ ∀f g s.
      f uniformly_continuous_on s ∧ g uniformly_continuous_on IMAGE f s ⇒
      g ∘ f uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_CMUL

⊢ ∀f c s.
      f uniformly_continuous_on s ⇒ (λx. c * f x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_ON_CLOSURE

⊢ ∀f s.
      f uniformly_continuous_on s ∧ f continuous_on closure s ⇒
      f uniformly_continuous_on closure s

UNIFORMLY_CONTINUOUS_ON_ADD

⊢ ∀f g s.
      f uniformly_continuous_on s ∧ g uniformly_continuous_on s ⇒
      (λx. f x + g x) uniformly_continuous_on s

UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS

⊢ ∀f s. f uniformly_continuous_on s ⇒ f continuous_on s

UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS

⊢ ∀f s.
      f uniformly_continuous_on s ⇒
      ∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)

UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE

⊢ ∀f s.
      f uniformly_continuous_on s ⇒
      ∃g.
          g uniformly_continuous_on closure s ∧ (∀x. x ∈ s ⇒ (g x = f x)) ∧
          ∀h.
              h continuous_on closure s ∧ (∀x. x ∈ s ⇒ (h x = f x)) ⇒
              ∀x. x ∈ closure s ⇒ (h x = g x)

UNIFORMLY_CAUCHY_IMP_UNIFORMLY_CONVERGENT

⊢ ∀P s l.
      (∀e. 0 < e ⇒ ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x ⇒ dist (s m x,s n x) < e) ∧
      (∀x. P x ⇒ ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ dist (s n x,l x) < e) ⇒
      ∀e. 0 < e ⇒ ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (s n x,l x) < e

UNIFORM_LIM_SUB

⊢ ∀net P f g l m.
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
      ∀e.
          0 < e ⇒
          eventually (λx. ∀n. P n ⇒ abs (f n x − g n x − (l n − m n)) < e) net

UNIFORM_LIM_BILINEAR

⊢ ∀net P h f g l m b1 b2.
      bilinear h ∧ eventually (λx. ∀n. P n ⇒ abs (l n) ≤ b1) net ∧
      eventually (λx. ∀n. P n ⇒ abs (m n) ≤ b2) net ∧
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
      ∀e.
          0 < e ⇒
          eventually
            (λx. ∀n. P n ⇒ abs (h (f n x) (g n x) − h (l n) (m n)) < e) net

UNIFORM_LIM_ADD

⊢ ∀net P f g l m.
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (f n x − l n) < e) net) ∧
      (∀e. 0 < e ⇒ eventually (λx. ∀n. P n ⇒ abs (g n x − m n) < e) net) ⇒
      ∀e.
          0 < e ⇒
          eventually (λx. ∀n. P n ⇒ abs (f n x + g n x − (l n + m n)) < e) net

UNCOUNTABLE_REAL

⊢ ¬COUNTABLE 𝕌(:real)

UNCOUNTABLE_OPEN

⊢ ∀s. open s ∧ s ≠ ∅ ⇒ ¬COUNTABLE s

UNCOUNTABLE_INTERVAL

⊢ (∀a b. interval (a,b) ≠ ∅ ⇒ ¬COUNTABLE (interval [(a,b)])) ∧
  ∀a b. interval (a,b) ≠ ∅ ⇒ ¬COUNTABLE (interval (a,b))

UNCOUNTABLE_EUCLIDEAN

⊢ ¬COUNTABLE 𝕌(:real)

UNBOUNDED_INTER_COBOUNDED

⊢ ∀s t. ¬bounded s ∧ bounded (𝕌(:real) DIFF t) ⇒ s ∩ t ≠ ∅

UNBOUNDED_HALFSPACE_COMPONENT_LT

⊢ ∀a. ¬bounded {x | x < a}

UNBOUNDED_HALFSPACE_COMPONENT_LE

⊢ ∀a. ¬bounded {x | x ≤ a}

UNBOUNDED_HALFSPACE_COMPONENT_GT

⊢ ∀a. ¬bounded {x | x > a}

UNBOUNDED_HALFSPACE_COMPONENT_GE

⊢ ∀a. ¬bounded {x | x ≥ a}

TRIVIAL_LIMIT_WITHIN

⊢ ∀a. trivial_limit (at a within s) ⇔ ¬(a limit_point_of s)

TRIVIAL_LIMIT_SEQUENTIALLY

⊢ ¬trivial_limit sequentially

TRIVIAL_LIMIT_AT_POSINFINITY

⊢ ¬trivial_limit at_posinfinity

TRIVIAL_LIMIT_AT_NEGINFINITY

⊢ ¬trivial_limit at_neginfinity

TRIVIAL_LIMIT_AT_INFINITY

⊢ ¬trivial_limit at_infinity

TRIVIAL_LIMIT_AT

⊢ ∀a. ¬trivial_limit (at a)

TRANSLATION_DIFF

⊢ ∀s t.
      IMAGE (λx. a + x) (s DIFF t) =
      IMAGE (λx. a + x) s DIFF IMAGE (λx. a + x) t

TRANSITIVE_STEPWISE_LT_EQ

⊢ ∀R.
      (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
      ((∀m n. m < n ⇒ R m n) ⇔ ∀n. R n (SUC n))

TRANSITIVE_STEPWISE_LT

⊢ ∀R.
      (∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (SUC n)) ⇒
      ∀m n. m < n ⇒ R m n

TRANSITIVE_STEPWISE_LE_EQ

⊢ ∀R.
      (∀x. R x x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ⇒
      ((∀m n. m ≤ n ⇒ R m n) ⇔ ∀n. R n (SUC n))

TRANSITIVE_STEPWISE_LE

⊢ ∀R.
      (∀x. R x x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧ (∀n. R n (SUC n)) ⇒
      ∀m n. m ≤ n ⇒ R m n

TOPSPACE_SUBTOPOLOGY

⊢ ∀top u. topspace (subtopology top u) = topspace top ∩ u

TOPSPACE_EUCLIDEAN_SUBTOPOLOGY

⊢ ∀s. topspace (subtopology euclidean s) = s

TOPSPACE_EUCLIDEAN

⊢ topspace euclidean = 𝕌(:real)

TENDSTO_LIM

⊢ ∀net f l. ¬trivial_limit net ∧ (f --> l) net ⇒ (lim net f = l)

SYMMETRIC_LINEAR_IMAGE

⊢ ∀f s. (∀x. x ∈ s ⇒ -x ∈ s) ∧ linear f ⇒ ∀x. x ∈ IMAGE f s ⇒ -x ∈ IMAGE f s

SYMMETRIC_INTERIOR

⊢ ∀s. (∀x. x ∈ s ⇒ -x ∈ s) ⇒ ∀x. x ∈ interior s ⇒ -x ∈ interior s

SYMMETRIC_CLOSURE

⊢ ∀s. (∀x. x ∈ s ⇒ -x ∈ s) ⇒ ∀x. x ∈ closure s ⇒ -x ∈ closure s

SURJECTIVE_IMAGE_EQ

⊢ ∀s t. (∀y. y ∈ t ⇒ ∃x. f x = y) ∧ (∀x. f x ∈ t ⇔ x ∈ s) ⇒ (IMAGE f s = t)

SUP_INSERT

⊢ ∀x s. bounded s ⇒ (sup (x INSERT s) = if s = ∅ then x else max x (sup s))

SUMS_SYM

⊢ ∀s t. {x + y | x ∈ s ∧ y ∈ t} = {y + x | y ∈ t ∧ x ∈ s}

SUMS_SUMMABLE

⊢ ∀f l s. (f sums l) s ⇒ summable s f

SUMS_REINDEX_GEN

⊢ ∀k a l s. ((λx. a (x + k)) sums l) s ⇔ (a sums l) (IMAGE (λi. i + k) s)

SUMS_REINDEX

⊢ ∀k a l n. ((λx. a (x + k)) sums l) (from n) ⇔ (a sums l) (from (n + k))

SUMS_OFFSET_REV

⊢ ∀f l m n.
      (f sums l) (from m) ∧ 0 < m ∧ n ≤ m ⇒
      (f sums l + sum (n .. m − 1) f) (from n)

SUMS_OFFSET

⊢ ∀f l m n.
      (f sums l) (from m) ∧ 0 < n ∧ m ≤ n ⇒
      (f sums l − sum (m .. n − 1) f) (from n)

SUMS_LIM

⊢ ∀f s. (f sums lim sequentially (λn. sum (s ∩ (0 .. n)) f)) s ⇔ summable s f

SUMS_INTERVALS

⊢ (∀a b c d.
       interval [(a,b)] ≠ ∅ ∧ interval [(c,d)] ≠ ∅ ⇒
       ({x + y | x ∈ interval [(a,b)] ∧ y ∈ interval [(c,d)]} =
        interval [(a + c,b + d)])) ∧
  ∀a b c d.
      interval (a,b) ≠ ∅ ∧ interval (c,d) ≠ ∅ ⇒
      ({x + y | x ∈ interval (a,b) ∧ y ∈ interval (c,d)} =
       interval (a + c,b + d))

SUMS_INFSUM

⊢ ∀f s. (f sums infsum s f) s ⇔ summable s f

SUMS_IFF

⊢ ∀f g k. (∀x. x ∈ k ⇒ (f x = g x)) ⇒ ((f sums l) k ⇔ (g sums l) k)

SUMS_FINITE_UNION

⊢ ∀f s t l. FINITE t ∧ (f sums l) s ⇒ (f sums l + sum (t DIFF s) f) (s ∪ t)

SUMS_FINITE_DIFF

⊢ ∀f t s l. t ⊆ s ∧ FINITE t ∧ (f sums l) s ⇒ (f sums l − sum t f) (s DIFF t)

SUMS_EQ

⊢ ∀f g k. (∀x. x ∈ k ⇒ (f x = g x)) ∧ (f sums l) k ⇒ (g sums l) k

SUMS_0

⊢ ∀f s. (∀n. n ∈ s ⇒ (f n = 0)) ⇒ (f sums 0) s

SUMMABLE_TRIVIAL

⊢ ∀f. summable ∅ f

SUMMABLE_SUBSET_ABSCONV

⊢ ∀x s t. summable s (λn. abs (x n)) ∧ t ⊆ s ⇒ summable t (λn. abs (x n))

SUMMABLE_SUBSET

⊢ ∀x s t. s ⊆ t ∧ summable t (λi. if i ∈ s then x i else 0) ⇒ summable s x

SUMMABLE_SUB

⊢ ∀x y s. summable s x ∧ summable s y ⇒ summable s (λn. x n − y n)

SUMMABLE_RESTRICT

⊢ ∀f k. summable 𝕌(:num) (λn. if n ∈ k then f n else 0) ⇔ summable k f

SUMMABLE_REINDEX

⊢ ∀k a n. summable (from n) (λx. a (x + k)) ⇔ summable (from (n + k)) a

SUMMABLE_REARRANGE

⊢ ∀x s p. summable s (λn. abs (x n)) ∧ p permutes s ⇒ summable s (x ∘ p)

SUMMABLE_NEG

⊢ ∀x s. summable s x ⇒ summable s (λn. -x n)

SUMMABLE_LINEAR

⊢ ∀f h s. summable s f ∧ linear h ⇒ summable s (λn. h (f n))

SUMMABLE_IMP_TOZERO

⊢ ∀f k. summable k f ⇒ ((λn. if n ∈ k then f n else 0) --> 0) sequentially

SUMMABLE_IMP_SUMS_BOUNDED

⊢ ∀f k. summable (from k) f ⇒ bounded {sum (k .. n) f | n ∈ 𝕌(:num)}

SUMMABLE_IMP_BOUNDED

⊢ ∀f k. summable k f ⇒ bounded (IMAGE f k)

SUMMABLE_IFF_EVENTUALLY

⊢ ∀f g k.
      (∃N. ∀n. N ≤ n ∧ n ∈ k ⇒ (f n = g n)) ⇒ (summable k f ⇔ summable k g)

SUMMABLE_IFF_COFINITE

⊢ ∀f s t. FINITE (s DIFF t ∪ (t DIFF s)) ⇒ (summable s f ⇔ summable t f)

SUMMABLE_IFF

⊢ ∀f g k. (∀x. x ∈ k ⇒ (f x = g x)) ⇒ (summable k f ⇔ summable k g)

SUMMABLE_FROM_ELSEWHERE

⊢ ∀f m n. summable (from m) f ⇒ summable (from n) f

SUMMABLE_EQ_EVENTUALLY

⊢ ∀f g k. (∃N. ∀n. N ≤ n ∧ n ∈ k ⇒ (f n = g n)) ∧ summable k f ⇒ summable k g

SUMMABLE_EQ_COFINITE

⊢ ∀f s t. FINITE (s DIFF t ∪ (t DIFF s)) ∧ summable s f ⇒ summable t f

SUMMABLE_EQ

⊢ ∀f g k. (∀x. x ∈ k ⇒ (f x = g x)) ∧ summable k f ⇒ summable k g

SUMMABLE_COMPONENT

⊢ ∀f s. summable s f ⇒ summable s (λi. f i)

SUMMABLE_COMPARISON

⊢ ∀f g s.
      summable s g ∧ (∃N. ∀n. n ≥ N ∧ n ∈ s ⇒ abs (f n) ≤ g n) ⇒ summable s f

SUMMABLE_CMUL

⊢ ∀s x c. summable s x ⇒ summable s (λn. c * x n)

SUMMABLE_CAUCHY

⊢ ∀f s.
      summable s f ⇔
      ∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ⇒ abs (sum (s ∩ (m .. n)) f) < e

SUMMABLE_BILINEAR_PARTIAL_PRE

⊢ ∀f g h l k.
      bilinear h ∧ ((λn. h (f (n + 1)) (g n)) --> l) sequentially ∧
      summable (from k) (λn. h (f (n + 1) − f n) (g n)) ⇒
      summable (from k) (λn. h (f n) (g n − g (n − 1)))

SUMMABLE_ADD

⊢ ∀x y s. summable s x ∧ summable s y ⇒ summable s (λn. x n + y n)

SUMMABLE_0

⊢ ∀s. summable s (λn. 0)

SUM_GP_MULTIPLIED

⊢ ∀x m n.
      m ≤ n ⇒ ((1 − x) * sum (m .. n) (λi. x pow i) = x pow m − x pow SUC n)

SUM_GP_BASIC

⊢ ∀x n. (1 − x) * sum (0 .. n) (λi. x pow i) = 1 − x pow SUC n

SUM_GP

⊢ ∀x m n.
      sum (m .. n) (λi. x pow i) =
      if n < m then 0
      else if x = 1 then &(n + 1 − m)
      else (x pow m − x pow SUC n) / (1 − x)

SUM_DIFF_LEMMA

⊢ ∀f k m n.
      m ≤ n ⇒
      (sum (k ∩ (0 .. n)) f − sum (k ∩ (0 .. m)) f = sum (k ∩ (m + 1 .. n)) f)

SUM_ABS_TRIANGLE

⊢ ∀s f b. FINITE s ∧ sum s (λa. abs (f a)) ≤ b ⇒ abs (sum s f) ≤ b

SUBTOPOLOGY_UNIV

⊢ ∀top. subtopology top 𝕌(:α) = top

SUBTOPOLOGY_TOPSPACE

⊢ ∀top. subtopology top (topspace top) = top

SUBTOPOLOGY_SUPERSET

⊢ ∀top s. topspace top ⊆ s ⇒ (subtopology top s = top)

SUBSPACE_UNIV

⊢ subspace 𝕌(:real)

SUBSPACE_UNION_CHAIN

⊢ ∀s t. subspace s ∧ subspace t ∧ subspace (s ∪ t) ⇒ s ⊆ t ∨ t ⊆ s

SUBSPACE_TRIVIAL

⊢ subspace {0}

SUBSPACE_TRANSLATION_SELF_EQ

⊢ ∀s a. subspace s ⇒ ((IMAGE (λx. a + x) s = s) ⇔ a ∈ s)

SUBSPACE_TRANSLATION_SELF

⊢ ∀s a. subspace s ∧ a ∈ s ⇒ (IMAGE (λx. a + x) s = s)

SUBSPACE_SUMS

⊢ ∀s t. subspace s ∧ subspace t ⇒ subspace {x + y | x ∈ s ∧ y ∈ t}

SUBSPACE_SUM

⊢ ∀s f t. subspace s ∧ FINITE t ∧ (∀x. x ∈ t ⇒ f x ∈ s) ⇒ sum t f ∈ s

SUBSPACE_SUBSTANDARD

⊢ subspace {x | x = 0}

SUBSPACE_SUB

⊢ ∀x y s. subspace s ∧ x ∈ s ∧ y ∈ s ⇒ x − y ∈ s

SUBSPACE_SPAN

⊢ ∀s. subspace (span s)

SUBSPACE_NEG

⊢ ∀x s. subspace s ∧ x ∈ s ⇒ -x ∈ s

SUBSPACE_MUL

⊢ ∀x c s. subspace s ∧ x ∈ s ⇒ c * x ∈ s

SUBSPACE_LINEAR_PREIMAGE

⊢ ∀f s. linear f ∧ subspace s ⇒ subspace {x | f x ∈ s}

SUBSPACE_LINEAR_IMAGE

⊢ ∀f s. linear f ∧ subspace s ⇒ subspace (IMAGE f s)

SUBSPACE_KERNEL

⊢ ∀f. linear f ⇒ subspace {x | f x = 0}

SUBSPACE_INTER

⊢ ∀s t. subspace s ∧ subspace t ⇒ subspace (s ∩ t)

SUBSPACE_IMP_NONEMPTY

⊢ ∀s. subspace s ⇒ s ≠ ∅

SUBSPACE_BOUNDED_EQ_TRIVIAL

⊢ ∀s. subspace s ⇒ (bounded s ⇔ (s = {0}))

SUBSPACE_BIGINTER

⊢ ∀f. (∀s. s ∈ f ⇒ subspace s) ⇒ subspace (BIGINTER f)

SUBSPACE_ADD

⊢ ∀x y s. subspace s ∧ x ∈ s ∧ y ∈ s ⇒ x + y ∈ s

SUBSPACE_0

⊢ subspace s ⇒ 0 ∈ s

SUBSET_INTERVAL_IMP

⊢ (a ≤ c ∧ d ≤ b ⇒ interval [(c,d)] ⊆ interval [(a,b)]) ∧
  (a < c ∧ d < b ⇒ interval [(c,d)] ⊆ interval (a,b)) ∧
  (a ≤ c ∧ d ≤ b ⇒ interval (c,d) ⊆ interval [(a,b)]) ∧
  (a ≤ c ∧ d ≤ b ⇒ interval (c,d) ⊆ interval (a,b))

SUBSET_INTERVAL

⊢ (interval [(c,d)] ⊆ interval [(a,b)] ⇔ c ≤ d ⇒ a ≤ c ∧ d ≤ b) ∧
  (interval [(c,d)] ⊆ interval (a,b) ⇔ c ≤ d ⇒ a < c ∧ d < b) ∧
  (interval (c,d) ⊆ interval [(a,b)] ⇔ c < d ⇒ a ≤ c ∧ d ≤ b) ∧
  (interval (c,d) ⊆ interval (a,b) ⇔ c < d ⇒ a ≤ c ∧ d ≤ b)

SUBSET_INTERIOR_EQ

⊢ ∀s. s ⊆ interior s ⇔ open s

SUBSET_INTERIOR

⊢ ∀s t. s ⊆ t ⇒ interior s ⊆ interior t

SUBSET_IMAGE

⊢ ∀f s t. s ⊆ IMAGE f t ⇔ ∃u. u ⊆ t ∧ (s = IMAGE f u)

SUBSET_CLOSURE

⊢ ∀s t. s ⊆ t ⇒ closure s ⊆ closure t

SUBSET_CBALL

⊢ ∀x d e. d ≤ e ⇒ cball (x,d) ⊆ cball (x,e)

SUBSET_BIGUNION

⊢ ∀f g. f ⊆ g ⇒ BIGUNION f ⊆ BIGUNION g

SUBSET_BALLS

⊢ (∀a a' r r'. ball (a,r) ⊆ ball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r ≤ 0) ∧
  (∀a a' r r'. ball (a,r) ⊆ cball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r ≤ 0) ∧
  (∀a a' r r'. cball (a,r) ⊆ ball (a',r') ⇔ dist (a,a') + r < r' ∨ r < 0) ∧
  ∀a a' r r'. cball (a,r) ⊆ cball (a',r') ⇔ dist (a,a') + r ≤ r' ∨ r < 0

SUBSET_BALL

⊢ ∀x d e. d ≤ e ⇒ ball (x,d) ⊆ ball (x,e)

SUBSET_ANTISYM_EQ

⊢ ∀s t. s ⊆ t ∧ t ⊆ s ⇔ (s = t)

SUBORDINATE_PARTITION_OF_UNITY

⊢ ∀c s.
      s ⊆ BIGUNION c ∧ (∀u. u ∈ c ⇒ open u) ∧
      (∀x. x ∈ s ⇒ ∃v. open v ∧ x ∈ v ∧ FINITE {u | u ∈ c ∧ u ∩ v ≠ ∅}) ⇒
      ∃f.
          (∀u. u ∈ c ⇒ f u continuous_on s ∧ ∀x. x ∈ s ⇒ 0 ≤ f u x) ∧
          (∀x u. u ∈ c ∧ x ∈ s ∧ x ∉ u ⇒ (f u x = 0)) ∧
          (∀x. x ∈ s ⇒ (sum c (λu. f u x) = 1)) ∧
          ∀x.
              x ∈ s ⇒
              ∃n.
                  open n ∧ x ∈ n ∧
                  FINITE {u | u ∈ c ∧ ¬∀x. x ∈ n ⇒ (f u x = 0)}

SPHERE_UNION_BALL

⊢ ∀a r. sphere (a,r) ∪ ball (a,r) = cball (a,r)

SPHERE_TRANSLATION

⊢ ∀a x r. sphere (a + x,r) = IMAGE (λy. a + y) (sphere (x,r))

SPHERE_SUBSET_CBALL

⊢ ∀x e. sphere (x,e) ⊆ cball (x,e)

SPHERE_SING

⊢ ∀x e. (e = 0) ⇒ (sphere (x,e) = {x})

SPHERE_LINEAR_IMAGE

⊢ ∀f x r.
      linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
      (sphere (f x,r) = IMAGE f (sphere (x,r)))

SPHERE_EQ_SING

⊢ ∀a r x. (sphere (a,r) = {x}) ⇔ (x = a) ∧ (r = 0)

SPHERE_EQ_EMPTY

⊢ ∀a r. (sphere (a,r) = ∅) ⇔ r < 0

SPHERE_EMPTY

⊢ ∀a r. r < 0 ⇒ (sphere (a,r) = ∅)

SPHERE

⊢ ∀a r. sphere (a,r) = if r < 0 then ∅ else {a − r; a + r}

SPANNING_SUBSET_INDEPENDENT

⊢ ∀s t. t ⊆ s ∧ independent s ∧ s ⊆ span t ⇒ (s = t)

SPAN_UNIV

⊢ span 𝕌(:real) = 𝕌(:real)

SPAN_UNION_SUBSET

⊢ ∀s t. span s ∪ span t ⊆ span (s ∪ t)

SPAN_UNION

⊢ ∀s t. span (s ∪ t) = {x + y | x ∈ span s ∧ y ∈ span t}

SPAN_TRANS

⊢ ∀x y s. x ∈ span s ∧ y ∈ span (x INSERT s) ⇒ y ∈ span s

SPAN_SUPERSET

⊢ ∀x. x ∈ s ⇒ x ∈ span s

SPAN_SUM

⊢ ∀s f t. FINITE t ∧ (∀x. x ∈ t ⇒ f x ∈ span s) ⇒ sum t f ∈ span s

SPAN_SUBSPACE

⊢ ∀b s. b ⊆ s ∧ s ⊆ span b ∧ subspace s ⇒ (span b = s)

SPAN_SUBSET_SUBSPACE

⊢ ∀s t. s ⊆ t ∧ subspace t ⇒ span s ⊆ t

SPAN_SUB

⊢ ∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x − y ∈ span s

SPAN_STDBASIS

⊢ span {i | 1 ≤ i ∧ i ≤ 1} = 𝕌(:real)

SPAN_SPAN

⊢ ∀s. span (span s) = span s

SPAN_NEG_EQ

⊢ ∀x s. -x ∈ span s ⇔ x ∈ span s

SPAN_NEG

⊢ ∀x s. x ∈ span s ⇒ -x ∈ span s

SPAN_MUL_EQ

⊢ ∀x c s. c ≠ 0 ⇒ (c * x ∈ span s ⇔ x ∈ span s)

SPAN_MUL

⊢ ∀x c s. x ∈ span s ⇒ c * x ∈ span s

SPAN_MONO

⊢ ∀s t. s ⊆ t ⇒ span s ⊆ span t

SPAN_LINEAR_IMAGE

⊢ ∀f s. linear f ⇒ (span (IMAGE f s) = IMAGE f (span s))

SPAN_INDUCT_ALT

⊢ ∀s h. h 0 ∧ (∀c x y. x ∈ s ∧ h y ⇒ h (c * x + y)) ⇒ ∀x. x ∈ span s ⇒ h x

SPAN_INDUCT

⊢ ∀s h. (∀x. x ∈ s ⇒ x ∈ h) ∧ subspace h ⇒ ∀x. x ∈ span s ⇒ h x

SPAN_INC

⊢ ∀s. s ⊆ span s

SPAN_EXPLICIT

⊢ ∀p. span p = {y | ∃s u. FINITE s ∧ s ⊆ p ∧ (sum s (λv. u v * v) = y)}

SPAN_EQ_SELF

⊢ ∀s. (span s = s) ⇔ subspace s

SPAN_EMPTY

⊢ span ∅ = {0}

SPAN_CLAUSES

⊢ (∀a s. a ∈ s ⇒ a ∈ span s) ∧ 0 ∈ span s ∧
  (∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x + y ∈ span s) ∧
  ∀x c s. x ∈ span s ⇒ c * x ∈ span s

SPAN_CARD_GE_DIM

⊢ ∀v b. v ⊆ span b ∧ FINITE b ⇒ dim v ≤ CARD b

SPAN_BREAKDOWN_EQ

⊢ ∀a s. x ∈ span (a INSERT s) ⇔ ∃k. x − k * a ∈ span s

SPAN_BREAKDOWN

⊢ ∀b s a. b ∈ s ∧ a ∈ span s ⇒ ∃k. a − k * b ∈ span (s DELETE b)

SPAN_ADD_EQ

⊢ ∀s x y. x ∈ span s ⇒ (x + y ∈ span s ⇔ y ∈ span s)

SPAN_ADD

⊢ ∀x y s. x ∈ span s ∧ y ∈ span s ⇒ x + y ∈ span s

SPAN_0

⊢ 0 ∈ span s

SIMPLE_IMAGE_GEN

⊢ ∀f P. {f x | P x} = IMAGE f {x | P x}

SETDIST_ZERO_STRONG

⊢ ∀s t. ¬DISJOINT (closure s) (closure t) ⇒ (setdist (s,t) = 0)

SETDIST_ZERO

⊢ ∀s t. ¬DISJOINT s t ⇒ (setdist (s,t) = 0)

SETDIST_UNIV

⊢ (∀s. setdist (s,𝕌(:real)) = 0) ∧ ∀t. setdist (𝕌(:real),t) = 0

SETDIST_UNIQUE

⊢ ∀s t a b d.
      a ∈ s ∧ b ∈ t ∧ (dist (a,b) = d) ∧
      (∀x y. x ∈ s ∧ y ∈ t ⇒ dist (a,b) ≤ dist (x,y)) ⇒
      (setdist (s,t) = d)

SETDIST_TRIANGLE

⊢ ∀s a t. setdist (s,t) ≤ setdist (s,{a}) + setdist ({a},t)

SETDIST_TRANSLATION

⊢ ∀a s t. setdist (IMAGE (λx. a + x) s,IMAGE (λx. a + x) t) = setdist (s,t)

SETDIST_SYM

⊢ ∀s t. setdist (s,t) = setdist (t,s)

SETDIST_SUBSETS_EQ

⊢ ∀s t s' t'.
      s' ⊆ s ∧ t' ⊆ t ∧
      (∀x y.
           x ∈ s ∧ y ∈ t ⇒
           ∃x' y'. x' ∈ s' ∧ y' ∈ t' ∧ dist (x',y') ≤ dist (x,y)) ⇒
      (setdist (s',t') = setdist (s,t))

SETDIST_SUBSET_RIGHT

⊢ ∀s t u. t ≠ ∅ ∧ t ⊆ u ⇒ setdist (s,u) ≤ setdist (s,t)

SETDIST_SUBSET_LEFT

⊢ ∀s t u. s ≠ ∅ ∧ s ⊆ t ⇒ setdist (t,u) ≤ setdist (s,u)

SETDIST_SINGS

⊢ ∀x y. setdist ({x},{y}) = dist (x,y)

SETDIST_SING_TRIANGLE

⊢ ∀s x y. abs (setdist ({x},s) − setdist ({y},s)) ≤ dist (x,y)

SETDIST_SING_LE_HAUSDIST

⊢ ∀s t x. bounded s ∧ bounded t ∧ x ∈ s ⇒ setdist ({x},t) ≤ hausdist (s,t)

SETDIST_SING_IN_SET

⊢ ∀x s. x ∈ s ⇒ (setdist ({x},s) = 0)

SETDIST_SING_FRONTIER_CASES

⊢ ∀s x. setdist ({x},s) = if x ∈ s then 0 else setdist ({x},frontier s)

SETDIST_SING_FRONTIER

⊢ ∀s x. x ∉ s ⇒ (setdist ({x},frontier s) = setdist ({x},s))

SETDIST_REFL

⊢ ∀s. setdist (s,s) = 0

SETDIST_POS_LE

⊢ ∀s t. 0 ≤ setdist (s,t)

SETDIST_LIPSCHITZ

⊢ ∀s t x y. abs (setdist ({x},s) − setdist ({y},s)) ≤ dist (x,y)

SETDIST_LINEAR_IMAGE

⊢ ∀f s t.
      linear f ∧ (∀x. abs (f x) = abs x) ⇒
      (setdist (IMAGE f s,IMAGE f t) = setdist (s,t))

SETDIST_LE_SING

⊢ ∀s t x. x ∈ s ⇒ setdist (s,t) ≤ setdist ({x},t)

SETDIST_LE_HAUSDIST

⊢ ∀s t. bounded s ∧ bounded t ⇒ setdist (s,t) ≤ hausdist (s,t)

SETDIST_LE_DIST

⊢ ∀s t x y. x ∈ s ∧ y ∈ t ⇒ setdist (s,t) ≤ dist (x,y)

SETDIST_HAUSDIST_TRIANGLE

⊢ ∀s t u.
      t ≠ ∅ ∧ bounded t ∧ bounded u ⇒
      setdist (s,u) ≤ setdist (s,t) + hausdist (t,u)

SETDIST_FRONTIERS

⊢ ∀s t.
      setdist (s,t) =
      if DISJOINT s t then setdist (frontier s,frontier t) else 0

SETDIST_FRONTIER

⊢ (∀s t. DISJOINT s t ⇒ (setdist (frontier s,t) = setdist (s,t))) ∧
  ∀s t. DISJOINT s t ⇒ (setdist (s,frontier t) = setdist (s,t))

SETDIST_EQ_0_SING

⊢ (∀s x. (setdist ({x},s) = 0) ⇔ (s = ∅) ∨ x ∈ closure s) ∧
  ∀s x. (setdist (s,{x}) = 0) ⇔ (s = ∅) ∨ x ∈ closure s

SETDIST_EQ_0_COMPACT_CLOSED

⊢ ∀s t.
      compact s ∧ closed t ⇒
      ((setdist (s,t) = 0) ⇔ (s = ∅) ∨ (t = ∅) ∨ s ∩ t ≠ ∅)

SETDIST_EQ_0_CLOSED_IN

⊢ ∀u s x.
      closed_in (subtopology euclidean u) s ∧ x ∈ u ⇒
      ((setdist ({x},s) = 0) ⇔ (s = ∅) ∨ x ∈ s)

SETDIST_EQ_0_CLOSED_COMPACT

⊢ ∀s t.
      closed s ∧ compact t ⇒
      ((setdist (s,t) = 0) ⇔ (s = ∅) ∨ (t = ∅) ∨ s ∩ t ≠ ∅)

SETDIST_EQ_0_CLOSED

⊢ ∀s x. closed s ⇒ ((setdist ({x},s) = 0) ⇔ (s = ∅) ∨ x ∈ s)

SETDIST_EQ_0_BOUNDED

⊢ ∀s t.
      bounded s ∨ bounded t ⇒
      ((setdist (s,t) = 0) ⇔ (s = ∅) ∨ (t = ∅) ∨ closure s ∩ closure t ≠ ∅)

SETDIST_EMPTY

⊢ (∀t. setdist (∅,t) = 0) ∧ ∀s. setdist (s,∅) = 0

SETDIST_DIFFERENCES

⊢ ∀s t. setdist (s,t) = setdist ({0},{x − y | x ∈ s ∧ y ∈ t})

SETDIST_COMPACT_CLOSED

⊢ ∀s t.
      compact s ∧ closed t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      ∃x y. x ∈ s ∧ y ∈ t ∧ (dist (x,y) = setdist (s,t))

SETDIST_CLOSURE

⊢ (∀s t. setdist (closure s,t) = setdist (s,t)) ∧
  ∀s t. setdist (s,closure t) = setdist (s,t)

SETDIST_CLOSEST_POINT

⊢ ∀a s. closed s ∧ s ≠ ∅ ⇒ (setdist ({a},s) = dist (a,closest_point s a))

SETDIST_CLOSED_COMPACT

⊢ ∀s t.
      closed s ∧ compact t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      ∃x y. x ∈ s ∧ y ∈ t ∧ (dist (x,y) = setdist (s,t))

SETDIST_BALLS

⊢ (∀a b r s.
       setdist (ball (a,r),ball (b,s)) =
       if r ≤ 0 ∨ s ≤ 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
  (∀a b r s.
       setdist (ball (a,r),cball (b,s)) =
       if r ≤ 0 ∨ s < 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
  (∀a b r s.
       setdist (cball (a,r),ball (b,s)) =
       if r < 0 ∨ s ≤ 0 then 0 else max 0 (dist (a,b) − (r + s))) ∧
  ∀a b r s.
      setdist (cball (a,r),cball (b,s)) =
      if r < 0 ∨ s < 0 then 0 else max 0 (dist (a,b) − (r + s))

SET_DIFF_FRONTIER

⊢ ∀s. s DIFF frontier s = interior s

SERIES_UNIQUE

⊢ ∀f l l' s. (f sums l) s ∧ (f sums l') s ⇒ (l = l')

SERIES_TRIVIAL

⊢ ∀f. (f sums 0) ∅

SERIES_TERMS_TOZERO

⊢ ∀f l n. (f sums l) (from n) ⇒ (f --> 0) sequentially

SERIES_SUM

⊢ ∀f l k s.
      FINITE s ∧ s ⊆ k ∧ (∀x. x ∉ s ⇒ (f x = 0)) ∧ (sum s f = l) ⇒
      (f sums l) k

SERIES_SUBSET

⊢ ∀x s t l. s ⊆ t ∧ ((λi. if i ∈ s then x i else 0) sums l) t ⇒ (x sums l) s

SERIES_SUB

⊢ ∀x x0 y y0 s.
      (x sums x0) s ∧ (y sums y0) s ⇒ ((λn. x n − y n) sums x0 − y0) s

SERIES_RESTRICT

⊢ ∀f k l. ((λn. if n ∈ k then f n else 0) sums l) 𝕌(:num) ⇔ (f sums l) k

SERIES_REARRANGE_EQ

⊢ ∀x s p l.
      summable s (λn. abs (x n)) ∧ p permutes s ⇒
      ((x ∘ p sums l) s ⇔ (x sums l) s)

SERIES_REARRANGE

⊢ ∀x s p l.
      summable s (λn. abs (x n)) ∧ p permutes s ∧ (x sums l) s ⇒
      (x ∘ p sums l) s

SERIES_RATIO

⊢ ∀c a s N.
      c < 1 ∧ (∀n. n ≥ N ⇒ abs (a (SUC n)) ≤ c * abs (a n)) ⇒ ∃l. (a sums l) s

SERIES_NEG

⊢ ∀x x0 s. (x sums x0) s ⇒ ((λn. -x n) sums -x0) s

SERIES_LINEAR

⊢ ∀f h l s. (f sums l) s ∧ linear h ⇒ ((λn. h (f n)) sums h l) s

SERIES_INJECTIVE_IMAGE_STRONG

⊢ ∀x s f.
      summable (IMAGE f s) (λn. abs (x n)) ∧
      (∀m n. m ∈ s ∧ n ∈ s ∧ (f m = f n) ⇒ (m = n)) ⇒
      ((λn. sum (IMAGE f s ∩ (0 .. n)) x − sum (s ∩ (0 .. n)) (x ∘ f)) --> 0)
        sequentially

SERIES_INJECTIVE_IMAGE

⊢ ∀x s f l.
      summable (IMAGE f s) (λn. abs (x n)) ∧
      (∀m n. m ∈ s ∧ n ∈ s ∧ (f m = f n) ⇒ (m = n)) ⇒
      ((x ∘ f sums l) s ⇔ (x sums l) (IMAGE f s))

SERIES_GOESTOZERO

⊢ ∀s x.
      summable s x ⇒
      ∀e. 0 < e ⇒ eventually (λn. n ∈ s ⇒ abs (x n) < e) sequentially

SERIES_FROM

⊢ ∀f l k. (f sums l) (from k) ⇔ ((λn. sum (k .. n) f) --> l) sequentially

SERIES_FINITE_SUPPORT

⊢ ∀f s k.
      FINITE (s ∩ k) ∧ (∀x. x ∈ k ∧ x ∉ s ⇒ (f x = 0)) ⇒
      (f sums sum (s ∩ k) f) k

SERIES_FINITE

⊢ ∀f s. FINITE s ⇒ (f sums sum s f) s

SERIES_DROP_POS

⊢ ∀f s a. (f sums a) s ∧ (∀x. x ∈ s ⇒ 0 ≤ f x) ⇒ 0 ≤ a

SERIES_DROP_LE

⊢ ∀f g s a b. (f sums a) s ∧ (g sums b) s ∧ (∀x. x ∈ s ⇒ f x ≤ g x) ⇒ a ≤ b

SERIES_DIRICHLET_BILINEAR

⊢ ∀f g h k m p l.
      bilinear h ∧ bounded {sum (m .. n) f | n ∈ 𝕌(:num)} ∧
      summable (from p) (λn. abs (g (n + 1) − g n)) ∧
      ((λn. h (g (n + 1)) (sum (1 .. n) f)) --> l) sequentially ⇒
      summable (from k) (λn. h (g n) (f n))

SERIES_DIRICHLET

⊢ ∀f g N k m.
      bounded {sum (m .. n) f | n ∈ 𝕌(:num)} ∧ (∀n. N ≤ n ⇒ g (n + 1) ≤ g n) ∧
      (g --> 0) sequentially ⇒
      summable (from k) (λn. g n * f n)

SERIES_DIFFS

⊢ ∀f k. (f --> 0) sequentially ⇒ ((λn. f n − f (n + 1)) sums f k) (from k)

SERIES_COMPONENT

⊢ ∀f s l. (f sums l) s ⇒ ((λi. f i) sums l) s

SERIES_COMPARISON_UNIFORM

⊢ ∀f g P s.
      (∃l. (g sums l) s) ∧ (∃N. ∀n x. N ≤ n ∧ n ∈ s ∧ P x ⇒ abs (f x n) ≤ g n) ⇒
      ∃l.
          ∀e.
              0 < e ⇒
              ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (sum (s ∩ (0 .. n)) (f x),l x) < e

SERIES_COMPARISON_BOUND

⊢ ∀f g s a.
      (g sums a) s ∧ (∀i. i ∈ s ⇒ abs (f i) ≤ g i) ⇒
      ∃l. (f sums l) s ∧ abs l ≤ a

SERIES_COMPARISON

⊢ ∀f g s.
      (∃l. (g sums l) s) ∧ (∃N. ∀n. n ≥ N ∧ n ∈ s ⇒ abs (f n) ≤ g n) ⇒
      ∃l. (f sums l) s

SERIES_CMUL

⊢ ∀x x0 c s. (x sums x0) s ⇒ ((λn. c * x n) sums c * x0) s

SERIES_CAUCHY_UNIFORM

⊢ ∀P f k.
      (∃l.
           ∀e.
               0 < e ⇒
               ∃N. ∀n x. N ≤ n ∧ P x ⇒ dist (sum (k ∩ (0 .. n)) (f x),l x) < e) ⇔
      ∀e. 0 < e ⇒ ∃N. ∀m n x. N ≤ m ∧ P x ⇒ abs (sum (k ∩ (m .. n)) (f x)) < e

SERIES_CAUCHY

⊢ ∀f s.
      (∃l. (f sums l) s) ⇔
      ∀e. 0 < e ⇒ ∃N. ∀m n. m ≥ N ⇒ abs (sum (s ∩ (m .. n)) f) < e

SERIES_BOUND

⊢ ∀f g s a b.
      (f sums a) s ∧ (g sums b) s ∧ (∀i. i ∈ s ⇒ abs (f i) ≤ g i) ⇒ abs a ≤ b

SERIES_ADD

⊢ ∀x x0 y y0 s.
      (x sums x0) s ∧ (y sums y0) s ⇒ ((λn. x n + y n) sums x0 + y0) s

SERIES_ABSCONV_IMP_CONV

⊢ ∀x k. summable k (λn. abs (x n)) ⇒ summable k x

SERIES_0

⊢ ∀s. ((λn. 0) sums 0) s

SEQUENTIALLY

⊢ ∀m n. netord sequentially m n ⇔ m ≥ n

SEQUENCE_UNIQUE_LIMPT

⊢ ∀f l l'.
      (f --> l) sequentially ∧ l' limit_point_of {y | ∃n. y = f n} ⇒ (l' = l)

SEQUENCE_INFINITE_LEMMA

⊢ ∀f l. (∀n. f n ≠ l) ∧ (f --> l) sequentially ⇒ INFINITE {y | (∃n. y = f n)}

SEQUENCE_CAUCHY_WLOG

⊢ ∀P s.
      (∀m n. P m ∧ P n ⇒ dist (s m,s n) < e) ⇔
      ∀m n. P m ∧ P n ∧ m ≤ n ⇒ dist (s m,s n) < e

SEQ_OFFSET_REV

⊢ ∀f l k. ((λi. f (i + k)) --> l) sequentially ⇒ (f --> l) sequentially

SEQ_OFFSET_NEG

⊢ ∀f l k. (f --> l) sequentially ⇒ ((λi. f (i − k)) --> l) sequentially

SEQ_OFFSET

⊢ ∀f l k. (f --> l) sequentially ⇒ ((λi. f (i + k)) --> l) sequentially

SEQ_HARMONIC_OFFSET

⊢ ∀a. ((λn. (&n + a)⁻¹) --> 0) sequentially

SEQ_HARMONIC

⊢ ((λn. (&n)⁻¹) --> 0) sequentially

SEPARATION_T2

⊢ ∀x y. x ≠ y ⇔ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∈ v ∧ (u ∩ v = ∅)

SEPARATION_T1

⊢ ∀x y. x ≠ y ⇔ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∉ u ∧ x ∉ v ∧ y ∈ v

SEPARATION_T0

⊢ ∀x y. x ≠ y ⇔ ∃u. open u ∧ (x ∈ u ⇎ y ∈ u)

SEPARATION_NORMAL_LOCAL

⊢ ∀s t u.
      closed_in (subtopology euclidean u) s ∧
      closed_in (subtopology euclidean u) t ∧ (s ∩ t = ∅) ⇒
      ∃s' t'.
          open_in (subtopology euclidean u) s' ∧
          open_in (subtopology euclidean u) t' ∧ s ⊆ s' ∧ t ⊆ t' ∧
          (s' ∩ t' = ∅)

SEPARATION_NORMAL_COMPACT

⊢ ∀s t.
      compact s ∧ closed t ∧ (s ∩ t = ∅) ⇒
      ∃u v.
          open u ∧ compact (closure u) ∧ open v ∧ s ⊆ u ∧ t ⊆ v ∧ (u ∩ v = ∅)

SEPARATION_NORMAL

⊢ ∀s t.
      closed s ∧ closed t ∧ (s ∩ t = ∅) ⇒
      ∃u v. open u ∧ open v ∧ s ⊆ u ∧ t ⊆ v ∧ (u ∩ v = ∅)

SEPARATION_HAUSDORFF

⊢ ∀x y. x ≠ y ⇒ ∃u v. open u ∧ open v ∧ x ∈ u ∧ y ∈ v ∧ (u ∩ v = ∅)

SEPARATION_CLOSURES

⊢ ∀s t.
      (s ∩ closure t = ∅) ∧ (t ∩ closure s = ∅) ⇒
      ∃u v. DISJOINT u v ∧ open u ∧ open v ∧ s ⊆ u ∧ t ⊆ v

SEPARATE_POINT_CLOSED

⊢ ∀s a. closed s ∧ a ∉ s ⇒ ∃d. 0 < d ∧ ∀x. x ∈ s ⇒ d ≤ dist (a,x)

SEPARATE_COMPACT_CLOSED

⊢ ∀s t.
      compact s ∧ closed t ∧ (s ∩ t = ∅) ⇒
      ∃d. 0 < d ∧ ∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)

SEPARATE_CLOSED_COMPACT

⊢ ∀s t.
      closed s ∧ compact t ∧ (s ∩ t = ∅) ⇒
      ∃d. 0 < d ∧ ∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)

SEGMENT_TRANSLATION

⊢ (∀c a b. segment [(c + a,c + b)] = IMAGE (λx. c + x) (segment [(a,b)])) ∧
  ∀c a b. segment (c + a,c + b) = IMAGE (λx. c + x) (segment (a,b))

SEGMENT_TO_POINT_EXISTS

⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ ∃b. b ∈ s ∧ (segment (a,b) ∩ s = ∅)

SEGMENT_TO_CLOSEST_POINT

⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ (segment (a,closest_point s a) ∩ s = ∅)

SEGMENT_SYM

⊢ (∀a b. segment [(a,b)] = segment [(b,a)]) ∧
  ∀a b. segment (a,b) = segment (b,a)

SEGMENT_SCALAR_MULTIPLE

⊢ (∀a b v. segment [(a * v,b * v)] = {x * v | a ≤ x ∧ x ≤ b ∨ b ≤ x ∧ x ≤ a}) ∧
  ∀a b v.
      v ≠ 0 ⇒
      (segment (a * v,b * v) = {x * v | a < x ∧ x < b ∨ b < x ∧ x < a})

SEGMENT_REFL

⊢ (∀a. segment [(a,a)] = {a}) ∧ ∀a. segment (a,a) = ∅

SEGMENT_OPEN_SUBSET_CLOSED

⊢ ∀a b. segment (a,b) ⊆ segment [(a,b)]

SEGMENT_CLOSED_OPEN

⊢ ∀a b. segment [(a,b)] = segment (a,b) ∪ {a; b}

segment

⊢ (segment [(a,b)] = {(1 − u) * a + u * b | 0 ≤ u ∧ u ≤ 1}) ∧
  (segment (a,b) = segment [(a,b)] DIFF {a; b})

SEGMENT

⊢ (∀a b.
       segment [(a,b)] = if a ≤ b then interval [(a,b)] else interval [(b,a)]) ∧
  ∀a b. segment (a,b) = if a ≤ b then interval (a,b) else interval (b,a)

REGULAR_OPEN_INTER

⊢ ∀s t.
      (interior (closure s) = s) ∧ (interior (closure t) = t) ⇒
      (interior (closure (s ∩ t)) = s ∩ t)

REGULAR_CLOSED_UNION

⊢ ∀s t.
      (closure (interior s) = s) ∧ (closure (interior t) = t) ⇒
      (closure (interior (s ∪ t)) = s ∪ t)

REGULAR_CLOSED_BIGUNION

⊢ ∀f.
      FINITE f ∧ (∀t. t ∈ f ⇒ (closure (interior t) = t)) ⇒
      (closure (interior (BIGUNION f)) = BIGUNION f)

REFLECT_INTERVAL

⊢ (∀a b. IMAGE (λx. -x) (interval [(a,b)]) = interval [(-b,-a)]) ∧
  ∀a b. IMAGE (λx. -x) (interval (a,b)) = interval (-b,-a)

REAL_WLOG_LT

⊢ (∀x. P x x) ∧ (∀x y. P x y ⇔ P y x) ∧ (∀x y. x < y ⇒ P x y) ⇒ ∀x y. P x y

REAL_WLOG_LE

⊢ (∀x y. P x y ⇔ P y x) ∧ (∀x y. x ≤ y ⇒ P x y) ⇒ ∀x y. P x y

REAL_SETDIST_LT_EXISTS

⊢ ∀s t b.
      s ≠ ∅ ∧ t ≠ ∅ ∧ setdist (s,t) < b ⇒ ∃x y. x ∈ s ∧ y ∈ t ∧ dist (x,y) < b

REAL_POW_LE_1

⊢ ∀n x. 1 ≤ x ⇒ 1 ≤ x pow n

REAL_POW_LBOUND

⊢ ∀x n. 0 ≤ x ⇒ 1 + &n * x ≤ (1 + x) pow n

REAL_POW_1_LE

⊢ ∀n x. 0 ≤ x ∧ x ≤ 1 ⇒ x pow n ≤ 1

REAL_OF_NUM_GE

⊢ ∀m n. &m ≥ &n ⇔ m ≥ n

REAL_LT_POW2

⊢ ∀n. 0 < 2 pow n

REAL_LT_MIN

⊢ ∀x y z. z < min x y ⇔ z < x ∧ z < y

REAL_LT_LCANCEL_IMP

⊢ ∀x y z. 0 < x ∧ x * y < x * z ⇒ y < z

REAL_LT_INV2

⊢ ∀x y. 0 < x ∧ x < y ⇒ y⁻¹ < x⁻¹

REAL_LT_INF_FINITE

⊢ ∀s a. FINITE s ∧ s ≠ ∅ ⇒ (a < inf s ⇔ ∀x. x ∈ s ⇒ a < x)

REAL_LT_HAUSDIST_POINT_EXISTS

⊢ ∀s t x d.
      bounded s ∧ bounded t ∧ t ≠ ∅ ∧ hausdist (s,t) < d ∧ x ∈ s ⇒
      ∃y. y ∈ t ∧ dist (x,y) < d

REAL_LT_AFFINITY

⊢ ∀m c x y. 0 < m ⇒ (y < m * x + c ⇔ m⁻¹ * y + -(c / m) < x)

REAL_LE_SQUARE_ABS

⊢ ∀x y. abs x ≤ abs y ⇔ x pow 2 ≤ y pow 2

REAL_LE_SETDIST_EQ

⊢ ∀d s t.
      d ≤ setdist (s,t) ⇔
      (∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)) ∧ ((s = ∅) ∨ (t = ∅) ⇒ d ≤ 0)

REAL_LE_SETDIST

⊢ ∀s t d.
      s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x y. x ∈ s ∧ y ∈ t ⇒ d ≤ dist (x,y)) ⇒
      d ≤ setdist (s,t)

REAL_LE_LMUL1

⊢ ∀x y z. 0 ≤ x ∧ y ≤ z ⇒ x * y ≤ x * z

REAL_LE_INV2

⊢ ∀x y. 0 < x ∧ x ≤ y ⇒ y⁻¹ ≤ x⁻¹

REAL_LE_INF_FINITE

⊢ ∀s a. FINITE s ∧ s ≠ ∅ ⇒ (a ≤ inf s ⇔ ∀x. x ∈ s ⇒ a ≤ x)

REAL_LE_HAUSDIST

⊢ ∀s t a b c z.
      s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
      (∀y. y ∈ t ⇒ setdist ({y},s) ≤ c) ∧
      (z ∈ s ∧ a ≤ setdist ({z},t) ∨ z ∈ t ∧ a ≤ setdist ({z},s)) ⇒
      a ≤ hausdist (s,t)

REAL_LE_BETWEEN

⊢ ∀a b. a ≤ b ⇔ ∃x. a ≤ x ∧ x ≤ b

REAL_LE_AFFINITY

⊢ ∀m c x y. 0 < m ⇒ (y ≤ m * x + c ⇔ m⁻¹ * y + -(c / m) ≤ x)

REAL_INV_LE_1

⊢ ∀x. 1 ≤ x ⇒ x⁻¹ ≤ 1

REAL_INV_1_LE

⊢ ∀x. 0 < x ∧ x ≤ 1 ⇒ 1 ≤ x⁻¹

REAL_INF_LT_FINITE

⊢ ∀s a. FINITE s ∧ s ≠ ∅ ⇒ (inf s < a ⇔ ∃x. x ∈ s ∧ x < a)

REAL_INF_LE_FINITE

⊢ ∀s a. FINITE s ∧ s ≠ ∅ ⇒ (inf s ≤ a ⇔ ∃x. x ∈ s ∧ x ≤ a)

REAL_HAUSDIST_LE_SUMS

⊢ ∀s t b.
      s ≠ ∅ ∧ t ≠ ∅ ∧ s ⊆ {y + z | y ∈ t ∧ z ∈ cball (0,b)} ∧
      t ⊆ {y + z | y ∈ s ∧ z ∈ cball (0,b)} ⇒
      hausdist (s,t) ≤ b

REAL_HAUSDIST_LE_EQ

⊢ ∀s t b.
      s ≠ ∅ ∧ t ≠ ∅ ∧ bounded s ∧ bounded t ⇒
      (hausdist (s,t) ≤ b ⇔
       (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧ ∀y. y ∈ t ⇒ setdist ({y},s) ≤ b)

REAL_HAUSDIST_LE

⊢ ∀s t b.
      s ≠ ∅ ∧ t ≠ ∅ ∧ (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
      (∀y. y ∈ t ⇒ setdist ({y},s) ≤ b) ⇒
      hausdist (s,t) ≤ b

REAL_HALF

⊢ (∀e. 0 < e / 2 ⇔ 0 < e) ∧ (∀e. e / 2 + e / 2 = e) ∧ ∀e. 2 * (e / 2) = e

REAL_EQ_SQUARE_ABS

⊢ ∀x y. (abs x = abs y) ⇔ (x pow 2 = y pow 2)

REAL_EQ_RINV

⊢ ∀x. (x = -x) ⇔ (x = 0)

REAL_EQ_LINV

⊢ ∀x. (-x = x) ⇔ (x = 0)

REAL_EQ_AFFINITY

⊢ ∀m c x y. m ≠ 0 ⇒ ((y = m * x + c) ⇔ (m⁻¹ * y + -(c / m) = x))

REAL_CONVEX_BOUND_LE

⊢ ∀x y a u v. x ≤ a ∧ y ≤ a ∧ 0 ≤ u ∧ 0 ≤ v ∧ (u + v = 1) ⇒ u * x + v * y ≤ a

REAL_CHOOSE_SIZE

⊢ ∀c. 0 ≤ c ⇒ ∃x. abs x = c

REAL_CHOOSE_DIST

⊢ ∀x e. 0 ≤ e ⇒ ∃y. dist (x,y) = e

REAL_BOUNDS_LT

⊢ ∀x k. -k < x ∧ x < k ⇔ abs x < k

REAL_ARCH_RDIV_EQ_0

⊢ ∀x c. 0 ≤ x ∧ 0 ≤ c ∧ (∀m. 0 < m ⇒ &m * x ≤ c) ⇒ (x = 0)

REAL_ARCH_POW_INV

⊢ ∀x y. 0 < y ∧ x < 1 ⇒ ∃n. x pow n < y

REAL_ARCH_POW2

⊢ ∀x. ∃n. x < 2 pow n

REAL_ARCH_POW

⊢ ∀x y. 1 < x ⇒ ∃n. y < x pow n

REAL_ARCH_INV

⊢ ∀e. 0 < e ⇔ ∃n. n ≠ 0 ∧ 0 < (&n)⁻¹ ∧ (&n)⁻¹ < e

REAL_AFFINITY_LT

⊢ ∀m c x y. 0 < m ⇒ (m * x + c < y ⇔ x < m⁻¹ * y + -(c / m))

REAL_AFFINITY_LE

⊢ ∀m c x y. 0 < m ⇒ (m * x + c ≤ y ⇔ x ≤ m⁻¹ * y + -(c / m))

REAL_AFFINITY_EQ

⊢ ∀m c x y. m ≠ 0 ⇒ ((m * x + c = y) ⇔ (x = m⁻¹ * y + -(c / m)))

QUOTIENT_MAP_RESTRICT

⊢ ∀f s t c.
      IMAGE f s ⊆ t ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ∧
      (open_in (subtopology euclidean t) c ∨
       closed_in (subtopology euclidean t) c) ⇒
      ∀u.
          u ⊆ c ⇒
          (open_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ c})
             {x | x ∈ {x | x ∈ s ∧ f x ∈ c} ∧ f x ∈ u} ⇔
           open_in (subtopology euclidean c) u)

QUOTIENT_MAP_OPEN_MAP_EQ

⊢ ∀f s t.
      IMAGE f s ⊆ t ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ⇒
      ((∀k.
            open_in (subtopology euclidean s) k ⇒
            open_in (subtopology euclidean t) (IMAGE f k)) ⇔
       ∀k.
           open_in (subtopology euclidean s) k ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ IMAGE f k})

QUOTIENT_MAP_OPEN_CLOSED

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀u.
            u ⊆ t ⇒
            (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
             open_in (subtopology euclidean t) u)) ⇔
       ∀u.
           u ⊆ t ⇒
           (closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            closed_in (subtopology euclidean t) u))

QUOTIENT_MAP_IMP_CONTINUOUS_OPEN

⊢ ∀f s t.
      IMAGE f s ⊆ t ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ⇒
      f continuous_on s

QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED

⊢ ∀f s t.
      IMAGE f s ⊆ t ∧
      (∀u.
           u ⊆ t ⇒
           (closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            closed_in (subtopology euclidean t) u)) ⇒
      f continuous_on s

QUOTIENT_MAP_FROM_SUBSET

⊢ ∀f s t u.
      f continuous_on t ∧ IMAGE f t ⊆ u ∧ s ⊆ t ∧ (IMAGE f s = u) ∧
      (∀v.
           v ⊆ u ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
            open_in (subtopology euclidean u) v)) ⇒
      ∀v.
          v ⊆ u ⇒
          (open_in (subtopology euclidean t) {x | x ∈ t ∧ f x ∈ v} ⇔
           open_in (subtopology euclidean u) v)

QUOTIENT_MAP_FROM_COMPOSITION

⊢ ∀f g s t u.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧ IMAGE g t ⊆ u ∧
      (∀v.
           v ⊆ u ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ (g ∘ f) x ∈ v} ⇔
            open_in (subtopology euclidean u) v)) ⇒
      ∀v.
          v ⊆ u ⇒
          (open_in (subtopology euclidean t) {x | x ∈ t ∧ g x ∈ v} ⇔
           open_in (subtopology euclidean u) v)

QUOTIENT_MAP_COMPOSE

⊢ ∀f g s t u.
      IMAGE f s ⊆ t ∧
      (∀v.
           v ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
            open_in (subtopology euclidean t) v)) ∧
      (∀v.
           v ⊆ u ⇒
           (open_in (subtopology euclidean t) {x | x ∈ t ∧ g x ∈ v} ⇔
            open_in (subtopology euclidean u) v)) ⇒
      ∀v.
          v ⊆ u ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ (g ∘ f) x ∈ v} ⇔
           open_in (subtopology euclidean u) v)

QUOTIENT_MAP_CLOSED_MAP_EQ

⊢ ∀f s t.
      IMAGE f s ⊆ t ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ⇒
      ((∀k.
            closed_in (subtopology euclidean s) k ⇒
            closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
       ∀k.
           closed_in (subtopology euclidean s) k ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ IMAGE f k})

QUASICOMPACT_OPEN_CLOSED

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀u.
            u ⊆ t ⇒
            open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇒
            open_in (subtopology euclidean t) u) ⇔
       ∀u.
           u ⊆ t ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇒
           closed_in (subtopology euclidean t) u)

PROPER_MAP_FROM_COMPOSITION_RIGHT

⊢ ∀f g s t u.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧ IMAGE g t ⊆ u ∧
      (∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}) ⇒
      ∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}

PROPER_MAP_FROM_COMPOSITION_LEFT

⊢ ∀f g s t u.
      f continuous_on s ∧ (IMAGE f s = t) ∧ g continuous_on t ∧
      IMAGE g t ⊆ u ∧
      (∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}) ⇒
      ∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ t ∧ g x ∈ k}

PROPER_MAP_FROM_COMPACT

⊢ ∀f s k.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ compact s ∧
      closed_in (subtopology euclidean t) k ⇒
      compact {x | x ∈ s ∧ f x ∈ k}

PROPER_MAP_COMPOSE

⊢ ∀f g s t u.
      IMAGE f s ⊆ t ∧
      (∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ∧
      (∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ t ∧ g x ∈ k}) ⇒
      ∀k. k ⊆ u ∧ compact k ⇒ compact {x | x ∈ s ∧ (g ∘ f) x ∈ k}

PROPER_MAP

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀k. k ⊆ t ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ⇔
       (∀k.
            closed_in (subtopology euclidean s) k ⇒
            closed_in (subtopology euclidean t) (IMAGE f k)) ∧
       ∀a. a ∈ t ⇒ compact {x | x ∈ s ∧ (f x = a)})

POWERSET_CLAUSES

⊢ ({s | s ⊆ ∅} = {∅}) ∧
  ∀a t.
      {s | s ⊆ a INSERT t} = {s | s ⊆ t} ∪ IMAGE (λs. a INSERT s) {s | s ⊆ t}

PERMUTES_INJECTIVE

⊢ ∀p s. p permutes s ⇒ ∀x y. (p x = p y) ⇔ (x = y)

PERMUTES_IMAGE

⊢ ∀p s. p permutes s ⇒ (IMAGE p s = s)

PASTING_LEMMA_EXISTS_CLOSED

⊢ ∀f t s k.
      FINITE k ∧ s ⊆ BIGUNION {t i | i ∈ k} ∧
      (∀i.
           i ∈ k ⇒
           closed_in (subtopology euclidean s) (t i) ∧ f i continuous_on t i) ∧
      (∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ (f i x = f j x)) ⇒
      ∃g. g continuous_on s ∧ ∀x i. i ∈ k ∧ x ∈ s ∩ t i ⇒ (g x = f i x)

PASTING_LEMMA_EXISTS

⊢ ∀f t s k.
      s ⊆ BIGUNION {t i | i ∈ k} ∧
      (∀i.
           i ∈ k ⇒
           open_in (subtopology euclidean s) (t i) ∧ f i continuous_on t i) ∧
      (∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ (f i x = f j x)) ⇒
      ∃g. g continuous_on s ∧ ∀x i. i ∈ k ∧ x ∈ s ∩ t i ⇒ (g x = f i x)

PASTING_LEMMA_CLOSED

⊢ ∀f g t s k.
      FINITE k ∧
      (∀i.
           i ∈ k ⇒
           closed_in (subtopology euclidean s) (t i) ∧ f i continuous_on t i) ∧
      (∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ (f i x = f j x)) ∧
      (∀x. x ∈ s ⇒ ∃j. j ∈ k ∧ x ∈ t j ∧ (g x = f j x)) ⇒
      g continuous_on s

PASTING_LEMMA

⊢ ∀f g t s k.
      (∀i.
           i ∈ k ⇒
           open_in (subtopology euclidean s) (t i) ∧ f i continuous_on t i) ∧
      (∀i j x. i ∈ k ∧ j ∈ k ∧ x ∈ s ∩ t i ∩ t j ⇒ (f i x = f j x)) ∧
      (∀x. x ∈ s ⇒ ∃j. j ∈ k ∧ x ∈ t j ∧ (g x = f j x)) ⇒
      g continuous_on s

PARTIAL_SUMS_DROP_LE_INFSUM

⊢ ∀f s n.
      (∀i. i ∈ s ⇒ 0 ≤ f i) ∧ summable s f ⇒ sum (s ∩ (0 .. n)) f ≤ infsum s f

PARTIAL_SUMS_COMPONENT_LE_INFSUM

⊢ ∀f s n.
      (∀i. i ∈ s ⇒ 0 ≤ f i) ∧ summable s f ⇒ sum (s ∩ (0 .. n)) f ≤ infsum s f

PAIRWISE_SING

⊢ ∀r x. pairwise r {x} ⇔ T

PAIRWISE_MONO

⊢ ∀r s t. pairwise r s ∧ t ⊆ s ⇒ pairwise r t

PAIRWISE_INSERT

⊢ ∀r x s.
      pairwise r (x INSERT s) ⇔
      (∀y. y ∈ s ∧ y ≠ x ⇒ r x y ∧ r y x) ∧ pairwise r s

PAIRWISE_IMAGE

⊢ ∀r f. pairwise r (IMAGE f s) ⇔ pairwise (λx y. f x ≠ f y ⇒ r (f x) (f y)) s

PAIRWISE_EMPTY

⊢ ∀r. pairwise r ∅ ⇔ T

PAIRWISE_DISJOINT_COMPONENTS

⊢ ∀u. pairwise DISJOINT (components u)

OPEN_UNIV

⊢ open 𝕌(:real)

OPEN_UNION_COMPACT_SUBSETS

⊢ ∀s.
      open s ⇒
      ∃f.
          (∀n. compact (f n)) ∧ (∀n. f n ⊆ s) ∧
          (∀n. f n ⊆ interior (f (n + 1))) ∧
          (BIGUNION {f n | n ∈ 𝕌(:num)} = s) ∧
          ∀k. compact k ∧ k ⊆ s ⇒ ∃N. ∀n. n ≥ N ⇒ k ⊆ f n

OPEN_UNION

⊢ ∀s t. open s ∧ open t ⇒ open (s ∪ t)

OPEN_TRANSLATION_EQ

⊢ ∀a s. open (IMAGE (λx. a + x) s) ⇔ open s

OPEN_TRANSLATION

⊢ ∀s a. open s ⇒ open (IMAGE (λx. a + x) s)

OPEN_SURJECTIVE_LINEAR_IMAGE

⊢ ∀f. linear f ∧ (∀y. ∃x. f x = y) ⇒ ∀s. open s ⇒ open (IMAGE f s)

OPEN_SUMS

⊢ ∀s t. open s ∨ open t ⇒ open {x + y | x ∈ s ∧ y ∈ t}

OPEN_SUBSET_INTERIOR

⊢ ∀s t. open s ⇒ (s ⊆ interior t ⇔ s ⊆ t)

OPEN_SUBSET

⊢ ∀s t. s ⊆ t ∧ open s ⇒ open_in (subtopology euclidean t) s

OPEN_SUB_OPEN

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃t. open t ∧ x ∈ t ∧ t ⊆ s

OPEN_SEGMENT_LINEAR_IMAGE

⊢ ∀f a b.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (segment (f a,f b) = IMAGE f (segment (a,b)))

OPEN_SEGMENT_ALT

⊢ ∀a b. a ≠ b ⇒ (segment (a,b) = {(1 − u) * a + u * b | 0 < u ∧ u < 1})

OPEN_SEGMENT

⊢ ∀a b. open (segment (a,b))

OPEN_SCALING

⊢ ∀s c. c ≠ 0 ∧ open s ⇒ open (IMAGE (λx. c * x) s)

OPEN_POSITIVE_ORTHANT

⊢ open {x | 0 < x}

OPEN_POSITIVE_MULTIPLES

⊢ ∀s. open s ⇒ open {c * x | 0 < c ∧ x ∈ s}

OPEN_OPEN_IN_TRANS

⊢ ∀s t. open s ∧ open t ∧ t ⊆ s ⇒ open_in (subtopology euclidean s) t

OPEN_NEGATIONS

⊢ ∀s. open s ⇒ open (IMAGE (λx. -x) s)

OPEN_MAP_RESTRICT

⊢ ∀f s t t'.
      (∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t) (IMAGE f u)) ∧ t' ⊆ t ⇒
      ∀u.
          open_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ t'}) u ⇒
          open_in (subtopology euclidean t') (IMAGE f u)

OPEN_MAP_INTERIORS

⊢ ∀f.
      (∀s. open s ⇒ open (IMAGE f s)) ⇔
      ∀s. IMAGE f (interior s) ⊆ interior (IMAGE f s)

OPEN_MAP_IMP_QUOTIENT_MAP

⊢ ∀f s.
      f continuous_on s ∧
      (∀t.
           open_in (subtopology euclidean s) t ⇒
           open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ⇒
      ∀t.
          t ⊆ IMAGE f s ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t} ⇔
           open_in (subtopology euclidean (IMAGE f s)) t)

OPEN_MAP_IMP_CLOSED_MAP

⊢ ∀f s t.
      (IMAGE f s = t) ∧
      (∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t) (IMAGE f u)) ∧
      (∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ IMAGE f u}) ⇒
      ∀u.
          closed_in (subtopology euclidean s) u ⇒
          closed_in (subtopology euclidean t) (IMAGE f u)

OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀u.
            open_in (subtopology euclidean s) u ⇒
            open_in (subtopology euclidean t) (IMAGE f u)) ⇔
       ∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t)
             {y | y ∈ t ∧ {x | x ∈ s ∧ (f x = y)} ⊆ u})

OPEN_MAP_FROM_COMPOSITION_SURJECTIVE

⊢ ∀f g s t u.
      f continuous_on s ∧ (IMAGE f s = t) ∧ IMAGE g t ⊆ u ∧
      (∀k.
           open_in (subtopology euclidean s) k ⇒
           open_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
      ∀k.
          open_in (subtopology euclidean t) k ⇒
          open_in (subtopology euclidean u) (IMAGE g k)

OPEN_MAP_FROM_COMPOSITION_INJECTIVE

⊢ ∀f g s t u.
      IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧ g continuous_on t ∧
      (∀x y. x ∈ t ∧ y ∈ t ∧ (g x = g y) ⇒ (x = y)) ∧
      (∀k.
           open_in (subtopology euclidean s) k ⇒
           open_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
      ∀k.
          open_in (subtopology euclidean s) k ⇒
          open_in (subtopology euclidean t) (IMAGE f k)

OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀k.
            open_in (subtopology euclidean s) k ⇒
            open_in (subtopology euclidean t) (IMAGE f k)) ⇔
       ∀u w.
           closed_in (subtopology euclidean s) u ∧ w ⊆ t ∧
           {x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
           ∃v.
               closed_in (subtopology euclidean t) v ∧ w ⊆ v ∧
               {x | x ∈ s ∧ f x ∈ v} ⊆ u)

OPEN_MAP_CLOSED_SUPERSET_PREIMAGE

⊢ ∀f s t u w.
      (∀k.
           open_in (subtopology euclidean s) k ⇒
           open_in (subtopology euclidean t) (IMAGE f k)) ∧
      closed_in (subtopology euclidean s) u ∧ w ⊆ t ∧
      {x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
      ∃v.
          closed_in (subtopology euclidean t) v ∧ w ⊆ v ∧
          {x | x ∈ s ∧ f x ∈ v} ⊆ u

OPEN_INTERVAL_RIGHT

⊢ ∀a. open {x | a < x}

OPEN_INTERVAL_MIDPOINT

⊢ ∀a b. interval (a,b) ≠ ∅ ⇒ 2⁻¹ * (a + b) ∈ interval (a,b)

OPEN_INTERVAL_LEMMA

⊢ ∀a b x. a < x ∧ x < b ⇒ ∃d. 0 < d ∧ ∀x'. abs (x' − x) < d ⇒ a < x' ∧ x' < b

OPEN_INTERVAL_LEFT

⊢ ∀b. open {x | x < b}

OPEN_INTERVAL_EQ

⊢ (∀a b. open (interval [(a,b)]) ⇔ (interval [(a,b)] = ∅)) ∧
  ∀a b. open (interval (a,b))

OPEN_INTERVAL

⊢ ∀a b. open (interval (a,b))

OPEN_INTERIOR

⊢ ∀s. open (interior s)

OPEN_INTER_CLOSURE_SUBSET

⊢ ∀s t. open s ⇒ s ∩ closure t ⊆ closure (s ∩ t)

OPEN_INTER_CLOSURE_EQ_EMPTY

⊢ ∀s t. open s ⇒ ((s ∩ closure t = ∅) ⇔ (s ∩ t = ∅))

OPEN_INTER

⊢ ∀s t. open s ∧ open t ⇒ open (s ∩ t)

OPEN_IN_TRANS_EQ

⊢ ∀s t.
      (∀u.
           open_in (subtopology euclidean t) u ⇒
           open_in (subtopology euclidean s) t) ⇔
      open_in (subtopology euclidean s) t

OPEN_IN_TRANS

⊢ ∀s t u.
      open_in (subtopology euclidean t) s ∧
      open_in (subtopology euclidean u) t ⇒
      open_in (subtopology euclidean u) s

OPEN_IN_SUBTOPOLOGY_UNION

⊢ ∀top s t u.
      open_in (subtopology top t) s ∧ open_in (subtopology top u) s ⇒
      open_in (subtopology top (t ∪ u)) s

OPEN_IN_SUBTOPOLOGY_REFL

⊢ ∀top u. open_in (subtopology top u) u ⇔ u ⊆ topspace top

OPEN_IN_SUBTOPOLOGY_INTER_SUBSET

⊢ ∀s u v.
      open_in (subtopology euclidean u) (u ∩ s) ∧ v ⊆ u ⇒
      open_in (subtopology euclidean v) (v ∩ s)

OPEN_IN_SUBTOPOLOGY_EMPTY

⊢ ∀top s. open_in (subtopology top ∅) s ⇔ (s = ∅)

OPEN_IN_SUBTOPOLOGY

⊢ ∀top u s. open_in (subtopology top u) s ⇔ ∃t. open_in top t ∧ (s = t ∩ u)

OPEN_IN_SUBSET_TRANS

⊢ ∀s t u.
      open_in (subtopology euclidean u) s ∧ s ⊆ t ∧ t ⊆ u ⇒
      open_in (subtopology euclidean t) s

OPEN_IN_SING

⊢ ∀s a. open_in (subtopology euclidean s) {a} ⇔ a ∈ s ∧ ¬(a limit_point_of s)

OPEN_IN_REFL

⊢ ∀s. open_in (subtopology euclidean s) s

OPEN_IN_OPEN_TRANS

⊢ ∀s t. open_in (subtopology euclidean t) s ∧ open t ⇒ open s

OPEN_IN_OPEN_INTER

⊢ ∀u s. open s ⇒ open_in (subtopology euclidean u) (u ∩ s)

OPEN_IN_OPEN_EQ

⊢ ∀s t. open s ⇒ (open_in (subtopology euclidean s) t ⇔ open t ∧ t ⊆ s)

OPEN_IN_OPEN

⊢ ∀s u. open_in (subtopology euclidean u) s ⇔ ∃t. open t ∧ (s = u ∩ t)

OPEN_IN_LOCALLY_COMPACT

⊢ ∀s t.
      locally compact s ⇒
      (open_in (subtopology euclidean s) t ⇔
       t ⊆ s ∧
       ∀k. compact k ∧ k ⊆ s ⇒ open_in (subtopology euclidean k) (k ∩ t))

OPEN_IN_INTER_OPEN

⊢ ∀s t u.
      open_in (subtopology euclidean u) s ∧ open t ⇒
      open_in (subtopology euclidean u) (s ∩ t)

OPEN_IN_IMP_SUBSET

⊢ ∀top s t. open_in (subtopology top s) t ⇒ t ⊆ s

OPEN_IN_DELETE

⊢ ∀u s a.
      open_in (subtopology euclidean u) s ⇒
      open_in (subtopology euclidean u) (s DELETE a)

OPEN_IN_CONTAINS_CBALL

⊢ ∀s t.
      open_in (subtopology euclidean t) s ⇔
      s ⊆ t ∧ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ cball (x,e) ∩ t ⊆ s

OPEN_IN_CONTAINS_BALL

⊢ ∀s t.
      open_in (subtopology euclidean t) s ⇔
      s ⊆ t ∧ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ball (x,e) ∩ t ⊆ s

OPEN_IN_CONNECTED_COMPONENT

⊢ ∀s x.
      FINITE {connected_component s x | x | x ∈ s} ⇒
      open_in (subtopology euclidean s) (connected_component s x)

OPEN_IN

⊢ ∀s. open s ⇔ open_in euclidean s

open_in

⊢ ∀u s.
      open_in (subtopology euclidean u) s ⇔
      s ⊆ u ∧ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. x' ∈ u ∧ dist (x',x) < e ⇒ x' ∈ s

OPEN_IMP_LOCALLY_COMPACT

⊢ ∀s. open s ⇒ locally compact s

OPEN_IMP_INFINITE

⊢ ∀s. open s ⇒ (s = ∅) ∨ INFINITE s

OPEN_HALFSPACE_LT

⊢ ∀a b. open {x | a * x < b}

OPEN_HALFSPACE_GT

⊢ ∀a b. open {x | a * x > b}

OPEN_HALFSPACE_COMPONENT_LT

⊢ ∀a. open {x | x < a}

OPEN_HALFSPACE_COMPONENT_GT

⊢ ∀a. open {x | x > a}

OPEN_EXISTS_IN

⊢ ∀P Q. (∀a. P a ⇒ open {x | Q a x}) ⇒ open {x | (∃a. P a ∧ Q a x)}

OPEN_EXISTS

⊢ ∀Q. (∀a. open {x | Q a x}) ⇒ open {x | (∃a. Q a x)}

OPEN_EMPTY

⊢ open ∅

OPEN_DIFF

⊢ ∀s t. open s ∧ closed t ⇒ open (s DIFF t)

OPEN_DELETE

⊢ ∀s x. open s ⇒ open (s DELETE x)

OPEN_CONTAINS_OPEN_INTERVAL

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval (a,b) ⊆ s

OPEN_CONTAINS_INTERVAL_OPEN_INTERVAL

⊢ (∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval [(a,b)] ⊆ s) ∧
  ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval (a,b) ⊆ s

OPEN_CONTAINS_INTERVAL

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃a b. x ∈ interval (a,b) ∧ interval [(a,b)] ⊆ s

OPEN_CONTAINS_CBALL_EQ

⊢ ∀s. open s ⇒ ∀x. x ∈ s ⇔ ∃e. 0 < e ∧ cball (x,e) ⊆ s

OPEN_CONTAINS_CBALL

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ cball (x,e) ⊆ s

OPEN_CONTAINS_BALL_EQ

⊢ ∀s. open s ⇒ ∀x. x ∈ s ⇔ ∃e. 0 < e ∧ ball (x,e) ⊆ s

OPEN_CONTAINS_BALL

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ball (x,e) ⊆ s

OPEN_CLOSED_INTERVAL_CONVEX

⊢ ∀a b x y e.
      x ∈ interval (a,b) ∧ y ∈ interval [(a,b)] ∧ 0 < e ∧ e ≤ 1 ⇒
      e * x + (1 − e) * y ∈ interval (a,b)

OPEN_CLOSED_INTERVAL

⊢ ∀a b. interval (a,b) = interval [(a,b)] DIFF {a; b}

OPEN_CLOSED

⊢ ∀s. open s ⇔ closed (𝕌(:real) DIFF s)

OPEN_BIJECTIVE_LINEAR_IMAGE_EQ

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ∧ (∀y. ∃x. f x = y) ⇒
      (open (IMAGE f s) ⇔ open s)

OPEN_BIGUNION

⊢ (∀s. s ∈ f ⇒ open s) ⇒ open (BIGUNION f)

OPEN_BIGINTER

⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ open t) ⇒ open (BIGINTER s)

OPEN_BALL

⊢ ∀x e. open (ball (x,e))

OPEN_AFFINITY

⊢ ∀s a c. open s ∧ c ≠ 0 ⇒ open (IMAGE (λx. a + c * x) s)

OPEN

⊢ ∀s. open s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀x'. abs (x' − x) < e ⇒ x' ∈ s

OLDNET

⊢ ∀n x y.
      netord n x x ∧ netord n y y ⇒
      ∃z. netord n z z ∧ ∀w. netord n w z ⇒ netord n w x ∧ netord n w y

NOWHERE_DENSE_UNION

⊢ ∀s t.
      (interior (closure (s ∪ t)) = ∅) ⇔
      (interior (closure s) = ∅) ∧ (interior (closure t) = ∅)

NOWHERE_DENSE_COUNTABLE_BIGUNION_CLOSED

⊢ ∀g.
      COUNTABLE g ∧ (∀s. s ∈ g ⇒ closed s ∧ (interior s = ∅)) ⇒
      (interior (BIGUNION g) = ∅)

NOWHERE_DENSE_COUNTABLE_BIGUNION

⊢ ∀g.
      COUNTABLE g ∧ (∀s. s ∈ g ⇒ (interior (closure s) = ∅)) ⇒
      (interior (BIGUNION g) = ∅)

NOWHERE_DENSE

⊢ ∀s.
      (interior (closure s) = ∅) ⇔
      ∀t. open t ∧ t ≠ ∅ ⇒ ∃u. open u ∧ u ≠ ∅ ∧ u ⊆ t ∧ (u ∩ s = ∅)

NOT_INTERVAL_UNIV

⊢ (∀a b. interval [(a,b)] ≠ 𝕌(:real)) ∧ ∀a b. interval (a,b) ≠ 𝕌(:real)

NOT_EVENTUALLY

⊢ ∀net p. (∀x. ¬p x) ∧ ¬trivial_limit net ⇒ ¬eventually p net

NOT_EQ

⊢ ∀a b. a ≠ b ⇔ a ≠ b

NOT_BOUNDED_UNIV

⊢ ¬bounded 𝕌(:real)

NONTRIVIAL_LIMIT_WITHIN

⊢ ∀net s. trivial_limit net ⇒ trivial_limit (net within s)

NO_LIMIT_POINT_IMP_CLOSED

⊢ ∀s. ¬(∃x. x limit_point_of s) ⇒ closed s

NETLIMIT_WITHIN_INTERIOR

⊢ ∀s x. x ∈ interior s ⇒ (netlimit (at x within s) = x)

NETLIMIT_WITHIN

⊢ ∀a s. ¬trivial_limit (at a within s) ⇒ (netlimit (at a within s) = a)

NETLIMIT_AT

⊢ ∀a. netlimit (at a) = a

net_tybij

⊢ (∀a. mk_net (netord a) = a) ∧
  ∀r.
      (∀x y. (∀z. r z x ⇒ r z y) ∨ ∀z. r z y ⇒ r z x) ⇔
      (netord (mk_net r) = r)

NET_DILEMMA

⊢ ∀net.
      (∃a. (∃x. netord net x a) ∧ ∀x. netord net x a ⇒ P x) ∧
      (∃b. (∃x. netord net x b) ∧ ∀x. netord net x b ⇒ Q x) ⇒
      ∃c. (∃x. netord net x c) ∧ ∀x. netord net x c ⇒ P x ∧ Q x

NET

⊢ ∀n x y. (∀z. netord n z x ⇒ netord n z y) ∨ ∀z. netord n z y ⇒ netord n z x

NEGATIONS_SPHERE

⊢ ∀r. IMAGE (λx. -x) (sphere (0,r)) = sphere (0,r)

NEGATIONS_CBALL

⊢ ∀r. IMAGE (λx. -x) (cball (0,r)) = cball (0,r)

NEGATIONS_BALL

⊢ ∀r. IMAGE (λx. -x) (ball (0,r)) = ball (0,r)

MUMFORD_LEMMA

⊢ ∀f s t y.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ locally compact s ∧ y ∈ t ∧
      compact {x | x ∈ s ∧ (f x = y)} ⇒
      ∃u v.
          open_in (subtopology euclidean s) u ∧
          open_in (subtopology euclidean t) v ∧ {x | x ∈ s ∧ (f x = y)} ⊆ u ∧
          y ∈ v ∧ IMAGE f u ⊆ v ∧
          ∀k. k ⊆ v ∧ compact k ⇒ compact {x | x ∈ u ∧ f x ∈ k}

MUL_CAUCHY_SCHWARZ_EQUAL

⊢ ∀x y. ((x * y) pow 2 = x * x * (y * y)) ⇔ collinear {0; x; y}

MONOTONE_SUBSEQUENCE

⊢ ∀s.
      ∃r.
          (∀m n. m < n ⇒ r m < r n) ∧
          ((∀m n. m ≤ n ⇒ s (r m) ≤ s (r n)) ∨ ∀m n. m ≤ n ⇒ s (r n) ≤ s (r m))

MONOTONE_BIGGER

⊢ ∀r. (∀m n. m < n ⇒ r m < r n) ⇒ ∀n. n ≤ r n

MIDPOINT_SYM

⊢ ∀a b. midpoint (a,b) = midpoint (b,a)

MIDPOINT_REFL

⊢ ∀x. midpoint (x,x) = x

MIDPOINT_LINEAR_IMAGE

⊢ ∀f a b. linear f ⇒ (midpoint (f a,f b) = f (midpoint (a,b)))

MIDPOINT_IN_SEGMENT

⊢ (∀a b. midpoint (a,b) ∈ segment [(a,b)]) ∧
  ∀a b. midpoint (a,b) ∈ segment (a,b) ⇔ a ≠ b

MIDPOINT_EQ_ENDPOINT

⊢ ∀a b.
      ((midpoint (a,b) = a) ⇔ (a = b)) ∧ ((midpoint (a,b) = b) ⇔ (a = b)) ∧
      ((a = midpoint (a,b)) ⇔ (a = b)) ∧ ((b = midpoint (a,b)) ⇔ (a = b))

MIDPOINT_COLLINEAR

⊢ ∀a b c.
      a ≠ c ⇒
      ((b = midpoint (a,c)) ⇔ collinear {a; b; c} ∧ (dist (a,b) = dist (b,c)))

MAXIMAL_INDEPENDENT_SUBSET_EXTEND

⊢ ∀s v. s ⊆ v ∧ independent s ⇒ ∃b. s ⊆ b ∧ b ⊆ v ∧ independent b ∧ v ⊆ span b

MAXIMAL_INDEPENDENT_SUBSET

⊢ ∀v. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b

MAPPING_CONNECTED_ONTO_SEGMENT

⊢ ∀s a b.
      connected s ∧ ¬(∃a. s ⊆ {a}) ⇒
      ∃f. f continuous_on s ∧ (IMAGE f s = segment [(a,b)])

LT_NZ

⊢ ∀n. 0 < n ⇔ n ≠ 0

LT_EXISTS

⊢ ∀m n. m < n ⇔ ∃d. n = m + SUC d

LOWER_HEMICONTINUOUS

⊢ ∀f t s.
      (∀x. x ∈ s ⇒ f x ⊆ t) ⇒
      ((∀u.
            closed_in (subtopology euclidean t) u ⇒
            closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ⊆ u}) ⇔
       ∀u.
           open_in (subtopology euclidean t) u ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∩ u ≠ ∅})

LOCALLY_TRANSLATION

⊢ ∀P.
      (∀a s. P (IMAGE (λx. a + x) s) ⇔ P s) ⇒
      ∀a s. locally P (IMAGE (λx. a + x) s) ⇔ locally P s

LOCALLY_SING

⊢ ∀P a. locally P {a} ⇔ P {a}

LOCALLY_OPEN_SUBSET

⊢ ∀P s t. locally P s ∧ open_in (subtopology euclidean s) t ⇒ locally P t

LOCALLY_OPEN_MAP_IMAGE

⊢ ∀P Q f s.
      f continuous_on s ∧
      (∀t.
           open_in (subtopology euclidean s) t ⇒
           open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ∧
      (∀t. t ⊆ s ∧ P t ⇒ Q (IMAGE f t)) ∧ locally P s ⇒
      locally Q (IMAGE f s)

LOCALLY_MONO

⊢ ∀P Q s. (∀t. P t ⇒ Q t) ∧ locally P s ⇒ locally Q s

LOCALLY_INTER

⊢ ∀P.
      (∀s t. P s ∧ P t ⇒ P (s ∩ t)) ⇒
      ∀s t. locally P s ∧ locally P t ⇒ locally P (s ∩ t)

LOCALLY_INJECTIVE_LINEAR_IMAGE

⊢ ∀P Q.
      (∀f s. linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒ (P (IMAGE f s) ⇔ Q s)) ⇒
      ∀f s.
          linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
          (locally P (IMAGE f s) ⇔ locally Q s)

LOCALLY_EMPTY

⊢ ∀P. locally P ∅

LOCALLY_DIFF_CLOSED

⊢ ∀P s t.
      locally P s ∧ closed_in (subtopology euclidean s) t ⇒
      locally P (s DIFF t)

LOCALLY_COMPACT_UNIV

⊢ locally compact 𝕌(:real)

LOCALLY_COMPACT_TRANSLATION_EQ

⊢ ∀a s. locally compact (IMAGE (λx. a + x) s) ⇔ locally compact s

LOCALLY_COMPACT_PROPER_IMAGE_EQ

⊢ ∀f s.
      f continuous_on s ∧
      (∀k. k ⊆ IMAGE f s ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ⇒
      (locally compact s ⇔ locally compact (IMAGE f s))

LOCALLY_COMPACT_PROPER_IMAGE

⊢ ∀f s.
      f continuous_on s ∧
      (∀k. k ⊆ IMAGE f s ∧ compact k ⇒ compact {x | x ∈ s ∧ f x ∈ k}) ∧
      locally compact s ⇒
      locally compact (IMAGE f s)

LOCALLY_COMPACT_OPEN_UNION

⊢ ∀s t.
      locally compact s ∧ locally compact t ∧
      open_in (subtopology euclidean (s ∪ t)) s ∧
      open_in (subtopology euclidean (s ∪ t)) t ⇒
      locally compact (s ∪ t)

LOCALLY_COMPACT_OPEN_INTER_CLOSURE

⊢ ∀s. locally compact s ⇒ ∃t. open t ∧ (s = t ∩ closure s)

LOCALLY_COMPACT_OPEN_IN

⊢ ∀s t.
      open_in (subtopology euclidean s) t ∧ locally compact s ⇒
      locally compact t

LOCALLY_COMPACT_INTER_CBALLS

⊢ ∀s.
      locally compact s ⇔
      ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀d. d ≤ e ⇒ closed (cball (x,d) ∩ s)

LOCALLY_COMPACT_INTER_CBALL

⊢ ∀s. locally compact s ⇔ ∀x. x ∈ s ⇒ ∃e. 0 < e ∧ closed (cball (x,e) ∩ s)

LOCALLY_COMPACT_INTER

⊢ ∀s t. locally compact s ∧ locally compact t ⇒ locally compact (s ∩ t)

LOCALLY_COMPACT_DELETE

⊢ ∀s a. locally compact s ⇒ locally compact (s DELETE a)

LOCALLY_COMPACT_COMPACT_SUBOPEN

⊢ ∀s.
      locally compact s ⇔
      ∀k t.
          k ⊆ s ∧ compact k ∧ open t ∧ k ⊆ t ⇒
          ∃u v.
              k ⊆ u ∧ u ⊆ v ∧ u ⊆ t ∧ v ⊆ s ∧
              open_in (subtopology euclidean s) u ∧ compact v

LOCALLY_COMPACT_COMPACT_ALT

⊢ ∀s.
      locally compact s ⇔
      ∀k.
          k ⊆ s ∧ compact k ⇒
          ∃u.
              k ⊆ u ∧ open_in (subtopology euclidean s) u ∧
              compact (closure u) ∧ closure u ⊆ s

LOCALLY_COMPACT_COMPACT

⊢ ∀s.
      locally compact s ⇔
      ∀k.
          k ⊆ s ∧ compact k ⇒
          ∃u v.
              k ⊆ u ∧ u ⊆ v ∧ v ⊆ s ∧ open_in (subtopology euclidean s) u ∧
              compact v

LOCALLY_COMPACT_CLOSED_UNION

⊢ ∀s t.
      locally compact s ∧ locally compact t ∧
      closed_in (subtopology euclidean (s ∪ t)) s ∧
      closed_in (subtopology euclidean (s ∪ t)) t ⇒
      locally compact (s ∪ t)

LOCALLY_COMPACT_CLOSED_INTER_OPEN

⊢ ∀s. locally compact s ⇔ ∃t u. closed t ∧ open u ∧ (s = t ∩ u)

LOCALLY_COMPACT_CLOSED_IN_OPEN

⊢ ∀s. locally compact s ⇒ ∃t. open t ∧ closed_in (subtopology euclidean t) s

LOCALLY_COMPACT_CLOSED_IN

⊢ ∀s t.
      closed_in (subtopology euclidean s) t ∧ locally compact s ⇒
      locally compact t

LOCALLY_COMPACT_ALT

⊢ ∀s.
      locally compact s ⇔
      ∀x.
          x ∈ s ⇒
          ∃u.
              x ∈ u ∧ open_in (subtopology euclidean s) u ∧
              compact (closure u) ∧ closure u ⊆ s

LOCALLY_COMPACT

⊢ ∀s.
      locally compact s ⇔
      ∀x.
          x ∈ s ⇒
          ∃u v.
              x ∈ u ∧ u ⊆ v ∧ v ⊆ s ∧ open_in (subtopology euclidean s) u ∧
              compact v

LOCALLY_CLOSED

⊢ ∀s. locally closed s ⇔ locally compact s

LINEAR_ZERO

⊢ linear (λx. 0)

LINEAR_UNIFORMLY_CONTINUOUS_ON

⊢ ∀f s. linear f ⇒ f uniformly_continuous_on s

LINEAR_SUM_MUL

⊢ ∀f s c v.
      linear f ∧ FINITE s ⇒
      (f (sum s (λi. c i * v i)) = sum s (λi. c i * f (v i)))

LINEAR_SUM

⊢ ∀f g s. linear f ∧ FINITE s ⇒ (f (sum s g) = sum s (f ∘ g))

LINEAR_SUB

⊢ ∀f x y. linear f ⇒ (f (x − y) = f x − f y)

LINEAR_SCALING

⊢ ∀c. linear (λx. c * x)

LINEAR_OPEN_MAPPING

⊢ ∀f g. linear f ∧ linear g ∧ (f ∘ g = I) ⇒ ∀s. open s ⇒ open (IMAGE f s)

LINEAR_NEGATION

⊢ linear (λx. -x)

LINEAR_NEG

⊢ ∀f x. linear f ⇒ (f (-x) = -f x)

LINEAR_MUL_COMPONENT

⊢ ∀f v. linear f ⇒ linear (λx. f x * v)

LINEAR_LIM_0

⊢ ∀f. linear f ⇒ (f --> 0) (at 0)

LINEAR_INVERTIBLE_BOUNDED_BELOW_POS

⊢ ∀f g.
      linear f ∧ linear g ∧ (g ∘ f = I) ⇒
      ∃B. 0 < B ∧ ∀x. B * abs x ≤ abs (f x)

LINEAR_INVERTIBLE_BOUNDED_BELOW

⊢ ∀f g. linear f ∧ linear g ∧ (g ∘ f = I) ⇒ ∃B. ∀x. B * abs x ≤ abs (f x)

LINEAR_INTERIOR_IMAGE_SUBSET

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      interior (IMAGE f s) ⊆ IMAGE f (interior s)

LINEAR_INJECTIVE_LEFT_INVERSE

⊢ ∀f.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      ∃g. linear g ∧ (g ∘ f = (λx. x))

LINEAR_INJECTIVE_IMP_SURJECTIVE

⊢ ∀f. linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒ ∀y. ∃x. f x = y

LINEAR_INJECTIVE_BOUNDED_BELOW_POS

⊢ ∀f.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      ∃B. 0 < B ∧ ∀x. abs x * B ≤ abs (f x)

LINEAR_INJECTIVE_0_SUBSPACE

⊢ ∀f s.
      linear f ∧ subspace s ⇒
      ((∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇔
       ∀x. x ∈ s ∧ (f x = 0) ⇒ (x = 0))

LINEAR_INDEPENDENT_EXTEND_LEMMA

⊢ ∀f b.
      FINITE b ⇒
      independent b ⇒
      ∃g.
          (∀x y. x ∈ span b ∧ y ∈ span b ⇒ (g (x + y) = g x + g y)) ∧
          (∀x c. x ∈ span b ⇒ (g (c * x) = c * g x)) ∧ ∀x. x ∈ b ⇒ (g x = f x)

LINEAR_INDEPENDENT_EXTEND

⊢ ∀f b. independent b ⇒ ∃g. linear g ∧ ∀x. x ∈ b ⇒ (g x = f x)

LINEAR_IMAGE_SUBSET_INTERIOR

⊢ ∀f s.
      linear f ∧ (∀y. ∃x. f x = y) ⇒
      IMAGE f (interior s) ⊆ interior (IMAGE f s)

LINEAR_ID

⊢ linear (λx. x)

LINEAR_EQ_STDBASIS

⊢ ∀f g. linear f ∧ linear g ∧ (∀i. 1 ≤ i ∧ i ≤ 1 ⇒ (f i = g i)) ⇒ (f = g)

LINEAR_EQ_0_SPAN

⊢ ∀f b. linear f ∧ (∀x. x ∈ b ⇒ (f x = 0)) ⇒ ∀x. x ∈ span b ⇒ (f x = 0)

LINEAR_EQ_0

⊢ ∀f b s.
      linear f ∧ s ⊆ span b ∧ (∀x. x ∈ b ⇒ (f x = 0)) ⇒ ∀x. x ∈ s ⇒ (f x = 0)

LINEAR_EQ

⊢ ∀f g b s.
      linear f ∧ linear g ∧ s ⊆ span b ∧ (∀x. x ∈ b ⇒ (f x = g x)) ⇒
      ∀x. x ∈ s ⇒ (f x = g x)

LINEAR_CONTINUOUS_WITHIN

⊢ ∀f s x. linear f ⇒ f continuous (at x within s)

LINEAR_CONTINUOUS_ON_COMPOSE

⊢ ∀f g s. f continuous_on s ∧ linear g ⇒ (λx. g (f x)) continuous_on s

LINEAR_CONTINUOUS_ON

⊢ ∀f s. linear f ⇒ f continuous_on s

LINEAR_CONTINUOUS_COMPOSE

⊢ ∀net f g. f continuous net ∧ linear g ⇒ (λx. g (f x)) continuous net

LINEAR_CONTINUOUS_AT

⊢ ∀f a. linear f ⇒ f continuous at a

LINEAR_COMPOSE_SUM

⊢ ∀f s. FINITE s ∧ (∀a. a ∈ s ⇒ linear (f a)) ⇒ linear (λx. sum s (λa. f a x))

LINEAR_COMPOSE_SUB

⊢ ∀f g. linear f ∧ linear g ⇒ linear (λx. f x − g x)

LINEAR_COMPOSE_NEG

⊢ ∀f. linear f ⇒ linear (λx. -f x)

LINEAR_COMPOSE_CMUL

⊢ ∀f c. linear f ⇒ linear (λx. c * f x)

LINEAR_COMPOSE_ADD

⊢ ∀f g. linear f ∧ linear g ⇒ linear (λx. f x + g x)

LINEAR_COMPOSE

⊢ ∀f g. linear f ∧ linear g ⇒ linear (g ∘ f)

LINEAR_CMUL

⊢ ∀f c x. linear f ⇒ (f (c * x) = c * f x)

LINEAR_BOUNDED_POS

⊢ ∀f. linear f ⇒ ∃B. 0 < B ∧ ∀x. abs (f x) ≤ B * abs x

LINEAR_BOUNDED

⊢ ∀f. linear f ⇒ ∃B. ∀x. abs (f x) ≤ B * abs x

LINEAR_ADD

⊢ ∀f x y. linear f ⇒ (f (x + y) = f x + f y)

LINEAR_0

⊢ ∀f. linear f ⇒ (f 0 = 0)

LIMPT_UNIV

⊢ ∀x. x limit_point_of 𝕌(:real)

LIMPT_SUBSET

⊢ ∀x s t. x limit_point_of s ∧ s ⊆ t ⇒ x limit_point_of t

LIMPT_SING

⊢ ∀x y. ¬(x limit_point_of {y})

LIMPT_SEQUENTIAL_INJ

⊢ ∀x s.
      x limit_point_of s ⇔
      ∃f.
          (∀n. f n ∈ s DELETE x) ∧ (∀m n. (f m = f n) ⇔ (m = n)) ∧
          (f --> x) sequentially

LIMPT_SEQUENTIAL

⊢ ∀x s.
      x limit_point_of s ⇔ ∃f. (∀n. f n ∈ s DELETE x) ∧ (f --> x) sequentially

LIMPT_OF_UNIV

⊢ ∀x. x limit_point_of 𝕌(:real)

LIMPT_OF_SEQUENCE_SUBSEQUENCE

⊢ ∀f l.
      l limit_point_of IMAGE f 𝕌(:num) ⇒
      ∃r. (∀m n. m < n ⇒ r m < r n) ∧ (f ∘ r --> l) sequentially

LIMPT_OF_OPEN_IN

⊢ ∀s t x.
      open_in (subtopology euclidean s) t ∧ x limit_point_of s ∧ x ∈ t ⇒
      x limit_point_of t

LIMPT_OF_OPEN

⊢ ∀s x. open s ∧ x ∈ s ⇒ x limit_point_of s

LIMPT_OF_LIMPTS

⊢ ∀x s. x limit_point_of {y | y limit_point_of s} ⇒ x limit_point_of s

LIMPT_OF_CLOSURE

⊢ ∀x s. x limit_point_of closure s ⇔ x limit_point_of s

LIMPT_INSERT

⊢ ∀s x y. x limit_point_of y INSERT s ⇔ x limit_point_of s

LIMPT_INJECTIVE_LINEAR_IMAGE_EQ

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (f x limit_point_of IMAGE f s ⇔ x limit_point_of s)

LIMPT_INFINITE_OPEN_BALL_CBALL

⊢ (∀s x. x limit_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ INFINITE (s ∩ t)) ∧
  (∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ ball (x,e))) ∧
  ∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ cball (x,e))

LIMPT_INFINITE_OPEN

⊢ ∀s x. x limit_point_of s ⇔ ∀t. x ∈ t ∧ open t ⇒ INFINITE (s ∩ t)

LIMPT_INFINITE_CBALL

⊢ ∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ cball (x,e))

LIMPT_INFINITE_BALL

⊢ ∀s x. x limit_point_of s ⇔ ∀e. 0 < e ⇒ INFINITE (s ∩ ball (x,e))

LIMPT_EMPTY

⊢ ∀x. ¬(x limit_point_of ∅)

LIMPT_BALL

⊢ ∀x y e. y limit_point_of ball (x,e) ⇔ 0 < e ∧ y ∈ cball (x,e)

LIMPT_APPROACHABLE_LE

⊢ ∀x s.
      x limit_point_of s ⇔ ∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ dist (x',x) ≤ e

LIMPT_APPROACHABLE

⊢ ∀x s.
      x limit_point_of s ⇔ ∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ dist (x',x) < e

LIMIT_POINT_UNION

⊢ ∀s t x. x limit_point_of s ∪ t ⇔ x limit_point_of s ∨ x limit_point_of t

LIMIT_POINT_FINITE

⊢ ∀s a. FINITE s ⇒ ¬(a limit_point_of s)

LIM_WITHIN_UNION

⊢ (f --> l) (at x within s ∪ t) ⇔
  (f --> l) (at x within s) ∧ (f --> l) (at x within t)

LIM_WITHIN_SUBSET

⊢ ∀f l a s. (f --> l) (at a within s) ∧ t ⊆ s ⇒ (f --> l) (at a within t)

LIM_WITHIN_OPEN

⊢ ∀f l a s. a ∈ s ∧ open s ⇒ ((f --> l) (at a within s) ⇔ (f --> l) (at a))

LIM_WITHIN_LE

⊢ ∀f l a s.
      (f --> l) (at a within s) ⇔
      ∀e.
          0 < e ⇒
          ∃d.
              0 < d ∧
              ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒ dist (f x,l) < e

LIM_WITHIN_INTERIOR

⊢ ∀f l s x. x ∈ interior s ⇒ ((f --> l) (at x within s) ⇔ (f --> l) (at x))

LIM_WITHIN_ID

⊢ ∀a s. ((λx. x) --> a) (at a within s)

LIM_WITHIN_EMPTY

⊢ ∀f l x. (f --> l) (at x within ∅)

LIM_WITHIN_CLOSED_TRIVIAL

⊢ ∀a s. closed s ∧ a ∉ s ⇒ trivial_limit (at a within s)

LIM_WITHIN

⊢ ∀f l a s.
      (f --> l) (at a within s) ⇔
      ∀e.
          0 < e ⇒
          ∃d.
              0 < d ∧
              ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) < d ⇒ dist (f x,l) < e

LIM_VMUL

⊢ ∀net c d v. (c --> d) net ⇒ ((λx. c x * v) --> (d * v)) net

LIM_UNIQUE

⊢ ∀net f l l'. ¬trivial_limit net ∧ (f --> l) net ∧ (f --> l') net ⇒ (l = l')

LIM_UNION_UNIV

⊢ ∀f x l s t.
      (f --> l) (at x within s) ∧ (f --> l) (at x within t) ∧
      (s ∪ t = 𝕌(:real)) ⇒
      (f --> l) (at x)

LIM_UNION

⊢ ∀f x l s t.
      (f --> l) (at x within s) ∧ (f --> l) (at x within t) ⇒
      (f --> l) (at x within s ∪ t)

LIM_TRANSFORM_WITHIN_SET_IMP

⊢ ∀f l a s t.
      eventually (λx. x ∈ t ⇒ x ∈ s) (at a) ∧ (f --> l) (at a within s) ⇒
      (f --> l) (at a within t)

LIM_TRANSFORM_WITHIN_SET

⊢ ∀f a s t.
      eventually (λx. x ∈ s ⇔ x ∈ t) (at a) ⇒
      ((f --> l) (at a within s) ⇔ (f --> l) (at a within t))

LIM_TRANSFORM_WITHIN_OPEN_IN

⊢ ∀f g s t a l.
      open_in (subtopology euclidean t) s ∧ a ∈ s ∧
      (∀x. x ∈ s ∧ x ≠ a ⇒ (f x = g x)) ∧ (f --> l) (at a within t) ⇒
      (g --> l) (at a within t)

LIM_TRANSFORM_WITHIN_OPEN

⊢ ∀f g s a l.
      open s ∧ a ∈ s ∧ (∀x. x ∈ s ∧ x ≠ a ⇒ (f x = g x)) ∧ (f --> l) (at a) ⇒
      (g --> l) (at a)

LIM_TRANSFORM_WITHIN

⊢ ∀f g x s d.
      0 < d ∧
      (∀x'. x' ∈ s ∧ 0 < dist (x',x) ∧ dist (x',x) < d ⇒ (f x' = g x')) ∧
      (f --> l) (at x within s) ⇒
      (g --> l) (at x within s)

LIM_TRANSFORM_EVENTUALLY

⊢ ∀net f g l. eventually (λx. f x = g x) net ∧ (f --> l) net ⇒ (g --> l) net

LIM_TRANSFORM_EQ

⊢ ∀net f g l. ((λx. f x − g x) --> 0) net ⇒ ((f --> l) net ⇔ (g --> l) net)

LIM_TRANSFORM_BOUND

⊢ ∀f g.
      eventually (λn. abs (f n) ≤ abs (g n)) net ∧ (g --> 0) net ⇒
      (f --> 0) net

LIM_TRANSFORM_AWAY_WITHIN

⊢ ∀f g a b s.
      a ≠ b ∧ (∀x. x ∈ s ∧ x ≠ a ∧ x ≠ b ⇒ (f x = g x)) ∧
      (f --> l) (at a within s) ⇒
      (g --> l) (at a within s)

LIM_TRANSFORM_AWAY_AT

⊢ ∀f g a b.
      a ≠ b ∧ (∀x. x ≠ a ∧ x ≠ b ⇒ (f x = g x)) ∧ (f --> l) (at a) ⇒
      (g --> l) (at a)

LIM_TRANSFORM_AT

⊢ ∀f g x d.
      0 < d ∧ (∀x'. 0 < dist (x',x) ∧ dist (x',x) < d ⇒ (f x' = g x')) ∧
      (f --> l) (at x) ⇒
      (g --> l) (at x)

LIM_TRANSFORM

⊢ ∀net f g l. ((λx. f x − g x) --> 0) net ∧ (f --> l) net ⇒ (g --> l) net

LIM_SUM

⊢ ∀net f l s.
      FINITE s ∧ (∀i. i ∈ s ⇒ (f i --> l i) net) ⇒
      ((λx. sum s (λi. f i x)) --> sum s l) net

LIM_SUBSEQUENCE

⊢ ∀s r l.
      (∀m n. m < n ⇒ r m < r n) ∧ (s --> l) sequentially ⇒
      (s ∘ r --> l) sequentially

LIM_SUB

⊢ ∀net f g l m.
      (f --> l) net ∧ (g --> m) net ⇒ ((λx. f x − g x) --> (l − m)) net

LIM_SEQUENTIALLY

⊢ ∀s l. (s --> l) sequentially ⇔ ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ dist (s n,l) < e

LIM_POSINFINITY_SEQUENTIALLY

⊢ ∀f l. (f --> l) at_posinfinity ⇒ ((λn. f (&n)) --> l) sequentially

LIM_NULL_SUM

⊢ ∀net f s.
      FINITE s ∧ (∀a. a ∈ s ⇒ ((λx. f x a) --> 0) net) ⇒
      ((λx. sum s (f x)) --> 0) net

LIM_NULL_SUB

⊢ ∀net f g. (f --> 0) net ∧ (g --> 0) net ⇒ ((λx. f x − g x) --> 0) net

LIM_NULL_COMPARISON

⊢ ∀net f g.
      eventually (λx. abs (f x) ≤ g x) net ∧ ((λx. g x) --> 0) net ⇒
      (f --> 0) net

LIM_NULL_CMUL_EQ

⊢ ∀net f c. c ≠ 0 ⇒ (((λx. c * f x) --> 0) net ⇔ (f --> 0) net)

LIM_NULL_CMUL_BOUNDED

⊢ ∀f g B.
      eventually (λa. (g a = 0) ∨ abs (f a) ≤ B) net ∧ (g --> 0) net ⇒
      ((λn. f n * g n) --> 0) net

LIM_NULL_CMUL

⊢ ∀net f c. (f --> 0) net ⇒ ((λx. c * f x) --> 0) net

LIM_NULL_ADD

⊢ ∀net f g. (f --> 0) net ∧ (g --> 0) net ⇒ ((λx. f x + g x) --> 0) net

LIM_NULL_ABS

⊢ ∀net f. (f --> 0) net ⇔ ((λx. abs (f x)) --> 0) net

LIM_NULL

⊢ ∀net f l. (f --> l) net ⇔ ((λx. f x − l) --> 0) net

LIM_NEG_EQ

⊢ ∀net f l. ((λx. -f x) --> -l) net ⇔ (f --> l) net

LIM_NEG

⊢ ∀net f l. (f --> l) net ⇒ ((λx. -f x) --> -l) net

LIM_MUL

⊢ ∀net f l c d.
      (c --> d) net ∧ (f --> l) net ⇒ ((λx. c x * f x) --> (d * l)) net

LIM_MIN

⊢ ∀net f g l m.
      (f --> l) net ∧ (g --> m) net ⇒ ((λx. min (f x) (g x)) --> min l m) net

LIM_MAX

⊢ ∀net f g l m.
      (f --> l) net ∧ (g --> m) net ⇒ ((λx. max (f x) (g x)) --> max l m) net

LIM_LINEAR

⊢ ∀net h f l. (f --> l) net ∧ linear h ⇒ ((λx. h (f x)) --> h l) net

LIM_LIFT_DOT

⊢ ∀f a. (f --> l) net ⇒ ((λy. a * f y) --> (a * l)) net

LIM_INV

⊢ ∀net f l. (f --> l) net ∧ l ≠ 0 ⇒ (realinv ∘ f --> l⁻¹) net

LIM_INFINITY_POSINFINITY

⊢ ∀f l. (f --> l) at_infinity ⇒ (f --> l) at_posinfinity

LIM_IN_CLOSED_SET

⊢ ∀net f s l.
      closed s ∧ eventually (λx. f x ∈ s) net ∧ ¬trivial_limit net ∧
      (f --> l) net ⇒
      l ∈ s

LIM_EVENTUALLY

⊢ ∀net f l. eventually (λx. f x = l) net ⇒ (f --> l) net

LIM_DROP_UBOUND

⊢ ∀net f l b.
      (f --> l) net ∧ ¬trivial_limit net ∧ eventually (λx. f x ≤ b) net ⇒
      l ≤ b

LIM_DROP_LE

⊢ ∀net f g l m.
      ¬trivial_limit net ∧ (f --> l) net ∧ (g --> m) net ∧
      eventually (λx. f x ≤ g x) net ⇒
      l ≤ m

LIM_DROP_LBOUND

⊢ ∀net f l b.
      (f --> l) net ∧ ¬trivial_limit net ∧ eventually (λx. b ≤ f x) net ⇒
      b ≤ l

LIM_CONTINUOUS_FUNCTION

⊢ ∀f net g l. f continuous at l ∧ (g --> l) net ⇒ ((λx. f (g x)) --> f l) net

LIM_CONST_EQ

⊢ ∀net c d. ((λx. c) --> d) net ⇔ trivial_limit net ∨ (c = d)

LIM_CONST

⊢ ∀net a. ((λx. a) --> a) net

LIM_CONG_WITHIN

⊢ (∀x. x ≠ a ⇒ (f x = g x)) ⇒
  (((λx. f x) --> l) (at a within s) ⇔ (g --> l) (at a within s))

LIM_CONG_AT

⊢ (∀x. x ≠ a ⇒ (f x = g x)) ⇒ (((λx. f x) --> l) (at a) ⇔ (g --> l) (at a))

LIM_COMPOSE_WITHIN

⊢ ∀net f g s y z.
      (f --> y) net ∧ eventually (λw. f w ∈ s ∧ ((f w = y) ⇒ (g y = z))) net ∧
      (g --> z) (at y within s) ⇒
      (g ∘ f --> z) net

LIM_COMPOSE_AT

⊢ ∀net f g y z.
      (f --> y) net ∧ eventually (λw. (f w = y) ⇒ (g y = z)) net ∧
      (g --> z) (at y) ⇒
      (g ∘ f --> z) net

LIM_COMPONENT_UBOUND

⊢ ∀net f l b k.
      ¬trivial_limit net ∧ (f --> l) net ∧ eventually (λx. f x ≤ b) net ⇒
      l ≤ b

LIM_COMPONENT_LE

⊢ ∀net f g l m.
      ¬trivial_limit net ∧ (f --> l) net ∧ (g --> m) net ∧
      eventually (λx. f x ≤ g x) net ⇒
      l ≤ m

LIM_COMPONENT_LBOUND

⊢ ∀net f l b.
      ¬trivial_limit net ∧ (f --> l) net ∧ eventually (λx. b ≤ f x) net ⇒
      b ≤ l

LIM_COMPONENT_EQ

⊢ ∀net f i l b.
      (f --> l) net ∧ ¬trivial_limit net ∧ eventually (λx. f x = b) net ⇒
      (l = b)

LIM_COMPONENT

⊢ ∀net f i l. (f --> l) net ⇒ ((λa. f a) --> l) net

LIM_CMUL_EQ

⊢ ∀net f l c. c ≠ 0 ⇒ (((λx. c * f x) --> (c * l)) net ⇔ (f --> l) net)

LIM_CMUL

⊢ ∀f l c. (f --> l) net ⇒ ((λx. c * f x) --> (c * l)) net

LIM_CASES_SEQUENTIALLY

⊢ ∀f g l m.
      (((λn. if m ≤ n then f n else g n) --> l) sequentially ⇔
       (f --> l) sequentially) ∧
      (((λn. if m < n then f n else g n) --> l) sequentially ⇔
       (f --> l) sequentially) ∧
      (((λn. if n ≤ m then f n else g n) --> l) sequentially ⇔
       (g --> l) sequentially) ∧
      (((λn. if n < m then f n else g n) --> l) sequentially ⇔
       (g --> l) sequentially)

LIM_CASES_FINITE_SEQUENTIALLY

⊢ ∀f g l.
      FINITE {n | P n} ⇒
      (((λn. if P n then f n else g n) --> l) sequentially ⇔
       (g --> l) sequentially)

LIM_CASES_COFINITE_SEQUENTIALLY

⊢ ∀f g l.
      FINITE {n | (¬P n)} ⇒
      (((λn. if P n then f n else g n) --> l) sequentially ⇔
       (f --> l) sequentially)

LIM_BILINEAR

⊢ ∀net h f g l m.
      (f --> l) net ∧ (g --> m) net ∧ bilinear h ⇒
      ((λx. h (f x) (g x)) --> h l m) net

LIM_AT_ZERO

⊢ ∀f l a. (f --> l) (at a) ⇔ ((λx. f (a + x)) --> l) (at 0)

LIM_AT_WITHIN

⊢ ∀f l a s. (f --> l) (at a) ⇒ (f --> l) (at a within s)

LIM_AT_POSINFINITY

⊢ ∀f l.
      (f --> l) at_posinfinity ⇔ ∀e. 0 < e ⇒ ∃b. ∀x. x ≥ b ⇒ dist (f x,l) < e

LIM_AT_NEGINFINITY

⊢ ∀f l.
      (f --> l) at_neginfinity ⇔ ∀e. 0 < e ⇒ ∃b. ∀x. x ≤ b ⇒ dist (f x,l) < e

LIM_AT_LE

⊢ ∀f l a.
      (f --> l) (at a) ⇔
      ∀e.
          0 < e ⇒
          ∃d. 0 < d ∧ ∀x. 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒ dist (f x,l) < e

LIM_AT_INFINITY_POS

⊢ ∀f l.
      (f --> l) at_infinity ⇔
      ∀e. 0 < e ⇒ ∃b. 0 < b ∧ ∀x. abs x ≥ b ⇒ dist (f x,l) < e

LIM_AT_INFINITY

⊢ ∀f l.
      (f --> l) at_infinity ⇔ ∀e. 0 < e ⇒ ∃b. ∀x. abs x ≥ b ⇒ dist (f x,l) < e

LIM_AT_ID

⊢ ∀a. ((λx. x) --> a) (at a)

LIM_AT

⊢ ∀f l a.
      (f --> l) (at a) ⇔
      ∀e.
          0 < e ⇒
          ∃d. 0 < d ∧ ∀x. 0 < dist (x,a) ∧ dist (x,a) < d ⇒ dist (f x,l) < e

LIM_ADD

⊢ ∀net f g l m.
      (f --> l) net ∧ (g --> m) net ⇒ ((λx. f x + g x) --> (l + m)) net

LIM_ABS_UBOUND

⊢ ∀net f l b.
      ¬trivial_limit net ∧ (f --> l) net ∧ eventually (λx. abs (f x) ≤ b) net ⇒
      abs l ≤ b

LIM_ABS_LBOUND

⊢ ∀net f l b.
      ¬trivial_limit net ∧ (f --> l) net ∧ eventually (λx. b ≤ abs (f x)) net ⇒
      b ≤ abs l

LIM_ABS

⊢ ∀net f l. (f --> l) net ⇒ ((λx. abs (f x)) --> abs l) net

LIM

⊢ (f --> l) net ⇔
  trivial_limit net ∨
  ∀e. 0 < e ⇒ ∃y. (∃x. netord net x y) ∧ ∀x. netord net x y ⇒ dist (f x,l) < e

LIFT_TO_QUOTIENT_SPACE_UNIQUE

⊢ ∀f g s t u.
      (IMAGE f s = t) ∧ (IMAGE g s = u) ∧
      (∀v.
           v ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
            open_in (subtopology euclidean t) v)) ∧
      (∀v.
           v ⊆ u ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ g x ∈ v} ⇔
            open_in (subtopology euclidean u) v)) ∧
      (∀x y. x ∈ s ∧ y ∈ s ⇒ ((f x = f y) ⇔ (g x = g y))) ⇒
      t homeomorphic u

LIFT_TO_QUOTIENT_SPACE

⊢ ∀f h s t u.
      (IMAGE f s = t) ∧
      (∀v.
           v ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
            open_in (subtopology euclidean t) v)) ∧ h continuous_on s ∧
      (IMAGE h s = u) ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (h x = h y)) ⇒
      ∃g. g continuous_on t ∧ (IMAGE g t = u) ∧ ∀x. x ∈ s ⇒ (h x = g (f x))

LEBESGUE_COVERING_LEMMA

⊢ ∀s c.
      compact s ∧ c ≠ ∅ ∧ s ⊆ BIGUNION c ∧ (∀b. b ∈ c ⇒ open b) ⇒
      ∃d. 0 < d ∧ ∀t. t ⊆ s ∧ diameter t ≤ d ⇒ ∃b. b ∈ c ∧ t ⊆ b

LE_EXISTS

⊢ ∀m n. m ≤ n ⇔ ∃d. n = m + d

LE_ADDR

⊢ ∀m n. n ≤ m + n

LE_ADD

⊢ ∀m n. m ≤ m + n

LE_1

⊢ (∀n. n ≠ 0 ⇒ 0 < n) ∧ (∀n. n ≠ 0 ⇒ 1 ≤ n) ∧ (∀n. 0 < n ⇒ n ≠ 0) ∧
  (∀n. 0 < n ⇒ 1 ≤ n) ∧ (∀n. 1 ≤ n ⇒ 0 < n) ∧ ∀n. 1 ≤ n ⇒ n ≠ 0

LAMBDA_PAIR

⊢ (λ(x,y). P x y) = (λp. P (FST p) (SND p))

JOINABLE_CONNECTED_COMPONENT_EQ

⊢ ∀s t x y.
      connected t ∧ t ⊆ s ∧ connected_component s x ∩ t ≠ ∅ ∧
      connected_component s y ∩ t ≠ ∅ ⇒
      (connected_component s x = connected_component s y)

JOINABLE_COMPONENTS_EQ

⊢ ∀s t c1 c2.
      connected t ∧ t ⊆ s ∧ c1 ∈ components s ∧ c2 ∈ components s ∧
      c1 ∩ t ≠ ∅ ∧ c2 ∩ t ≠ ∅ ⇒
      (c1 = c2)

ISTOPLOGY_SUBTOPOLOGY

⊢ ∀top u. istopology {s ∩ u | open_in top s}

ISOMETRY_ON_IMP_CONTINUOUS_ON

⊢ ∀f.
      (∀x y. x ∈ s ∧ y ∈ s ⇒ (dist (f x,f y) = dist (x,y))) ⇒
      f continuous_on s

ISOMETRY_IMP_OPEN_MAP

⊢ ∀f s t u.
      (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ⇒ (dist (f x,f y) = dist (x,y))) ∧
      open_in (subtopology euclidean s) u ⇒
      open_in (subtopology euclidean t) (IMAGE f u)

ISOMETRY_IMP_HOMEOMORPHISM_COMPACT

⊢ ∀f s.
      compact s ∧ IMAGE f s ⊆ s ∧
      (∀x y. x ∈ s ∧ y ∈ s ⇒ (dist (f x,f y) = dist (x,y))) ⇒
      ∃g. homeomorphism (s,s) (f,g)

ISOMETRY_IMP_EMBEDDING

⊢ ∀f s t.
      (IMAGE f s = t) ∧ (∀x y. x ∈ s ∧ y ∈ s ⇒ (dist (f x,f y) = dist (x,y))) ⇒
      ∃g. homeomorphism (s,t) (f,g)

IS_INTERVAL_UNIV

⊢ is_interval 𝕌(:real)

IS_INTERVAL_SING

⊢ ∀a. is_interval {a}

IS_INTERVAL_SCALING_EQ

⊢ ∀s c. is_interval (IMAGE (λx. c * x) s) ⇔ (c = 0) ∨ is_interval s

IS_INTERVAL_SCALING

⊢ ∀s c. is_interval s ⇒ is_interval (IMAGE (λx. c * x) s)

IS_INTERVAL_POINTWISE

⊢ ∀s x. is_interval s ⇒ (∃a. a ∈ s ∧ (a = x)) ⇒ x ∈ s

IS_INTERVAL_INTERVAL

⊢ ∀a b. is_interval (interval (a,b)) ∧ is_interval (interval [(a,b)])

IS_INTERVAL_INTER

⊢ ∀s t. is_interval s ∧ is_interval t ⇒ is_interval (s ∩ t)

IS_INTERVAL_IMP_LOCALLY_COMPACT

⊢ ∀s. is_interval s ⇒ locally compact s

IS_INTERVAL_EMPTY

⊢ is_interval ∅

IS_INTERVAL_COMPACT

⊢ ∀s. is_interval s ∧ compact s ⇔ ∃a b. s = interval [(a,b)]

IS_INTERVAL_CASES

⊢ ∀s.
      is_interval s ⇔
      (s = ∅) ∨ (s = 𝕌(:real)) ∨ (∃a. s = {x | a < x}) ∨
      (∃a. s = {x | a ≤ x}) ∨ (∃b. s = {x | x ≤ b}) ∨ (∃b. s = {x | x < b}) ∨
      (∃a b. s = {x | a < x ∧ x < b}) ∨ (∃a b. s = {x | a < x ∧ x ≤ b}) ∨
      (∃a b. s = {x | a ≤ x ∧ x < b}) ∨ ∃a b. s = {x | a ≤ x ∧ x ≤ b}

IS_INTERVAL

⊢ ∀s. is_interval s ⇔ ∀a b x. a ∈ s ∧ b ∈ s ∧ a ≤ x ∧ x ≤ b ⇒ x ∈ s

INTERVAL_TRANSLATION

⊢ (∀c a b. interval [(c + a,c + b)] = IMAGE (λx. c + x) (interval [(a,b)])) ∧
  ∀c a b. interval (c + a,c + b) = IMAGE (λx. c + x) (interval (a,b))

INTERVAL_SUBSET_IS_INTERVAL

⊢ ∀s a b.
      is_interval s ⇒
      (interval [(a,b)] ⊆ s ⇔ (interval [(a,b)] = ∅) ∨ a ∈ s ∧ b ∈ s)

INTERVAL_SING

⊢ (interval [(a,a)] = {a}) ∧ (interval (a,a) = ∅)

INTERVAL_OPEN_SUBSET_CLOSED

⊢ ∀a b. interval (a,b) ⊆ interval [(a,b)]

INTERVAL_NE_EMPTY

⊢ (interval [(a,b)] ≠ ∅ ⇔ a ≤ b) ∧ (interval (a,b) ≠ ∅ ⇔ a < b)

INTERVAL_IMAGE_STRETCH_INTERVAL

⊢ ∀a b m.
      ∃u v. IMAGE (λx. @f. f = m 1 * x) (interval [(a,b)]) = interval [(u,v)]

INTERVAL_EQ_EMPTY

⊢ ∀a b. (b < a ⇔ (interval [(a,b)] = ∅)) ∧ (b ≤ a ⇔ (interval (a,b) = ∅))

INTERVAL_CONTAINS_COMPACT_NEIGHBOURHOOD

⊢ ∀s x.
      is_interval s ∧ x ∈ s ⇒
      ∃a b d.
          0 < d ∧ x ∈ interval [(a,b)] ∧ interval [(a,b)] ⊆ s ∧
          ball (x,d) ∩ s ⊆ interval [(a,b)]

INTERVAL_CASES

⊢ ∀x. x ∈ interval [(a,b)] ⇒ x ∈ interval (a,b) ∨ (x = a) ∨ (x = b)

interval

⊢ (interval (a,b) = {x | a < x ∧ x < b}) ∧
  (interval [(a,b)] = {x | a ≤ x ∧ x ≤ b})

INTERVAL

⊢ (∀a b.
       interval [(a,b)] =
       if a ≤ b then cball (midpoint (a,b),dist (a,b) / 2) else ∅) ∧
  ∀a b.
      interval (a,b) =
      if a < b then ball (midpoint (a,b),dist (a,b) / 2) else ∅

INTERIOR_UNIV

⊢ interior 𝕌(:real) = 𝕌(:real)

INTERIOR_UNIQUE

⊢ ∀s t. t ⊆ s ∧ open t ∧ (∀t'. t' ⊆ s ∧ open t' ⇒ t' ⊆ t) ⇒ (interior s = t)

INTERIOR_UNIONS_OPEN_SUBSETS

⊢ ∀s. BIGUNION {t | open t ∧ t ⊆ s} = interior s

INTERIOR_UNION_EQ_EMPTY

⊢ ∀s t.
      closed s ∨ closed t ⇒
      ((interior (s ∪ t) = ∅) ⇔ (interior s = ∅) ∧ (interior t = ∅))

INTERIOR_TRANSLATION

⊢ ∀a s. interior (IMAGE (λx. a + x) s) = IMAGE (λx. a + x) (interior s)

INTERIOR_SUBSET

⊢ ∀s. interior s ⊆ s

INTERIOR_STANDARD_HYPERPLANE

⊢ ∀a. interior {x | x = a} = ∅

INTERIOR_SING

⊢ ∀a. interior {a} = ∅

INTERIOR_OPEN

⊢ ∀s. open s ⇒ (interior s = s)

INTERIOR_NEGATIONS

⊢ ∀s. interior (IMAGE (λx. -x) s) = IMAGE (λx. -x) (interior s)

INTERIOR_MAXIMAL_EQ

⊢ ∀s t. open s ⇒ (s ⊆ interior t ⇔ s ⊆ t)

INTERIOR_MAXIMAL

⊢ ∀s t. t ⊆ s ∧ open t ⇒ t ⊆ interior s

INTERIOR_LIMIT_POINT

⊢ ∀s x. x ∈ interior s ⇒ x limit_point_of s

INTERIOR_INTERVAL

⊢ (∀a b. interior (interval [(a,b)]) = interval (a,b)) ∧
  ∀a b. interior (interval (a,b)) = interval (a,b)

INTERIOR_INTERIOR

⊢ ∀s. interior (interior s) = interior s

INTERIOR_INTER

⊢ ∀s t. interior (s ∩ t) = interior s ∩ interior t

INTERIOR_INJECTIVE_LINEAR_IMAGE

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (interior (IMAGE f s) = IMAGE f (interior s))

INTERIOR_IMAGE_SUBSET

⊢ ∀f s.
      (∀x. f continuous at x) ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      interior (IMAGE f s) ⊆ IMAGE f (interior s)

INTERIOR_HYPERPLANE

⊢ ∀a b. a ≠ 0 ⇒ (interior {x | a * x = b} = ∅)

INTERIOR_HALFSPACE_LE

⊢ ∀a b. a ≠ 0 ⇒ (interior {x | a * x ≤ b} = {x | a * x < b})

INTERIOR_HALFSPACE_GE

⊢ ∀a b. a ≠ 0 ⇒ (interior {x | a * x ≥ b} = {x | a * x > b})

INTERIOR_HALFSPACE_COMPONENT_LE

⊢ ∀a. interior {x | x ≤ a} = {x | x < a}

INTERIOR_HALFSPACE_COMPONENT_GE

⊢ ∀a. interior {x | x ≥ a} = {x | x > a}

INTERIOR_FRONTIER_EMPTY

⊢ ∀s. open s ∨ closed s ⇒ (interior (frontier s) = ∅)

INTERIOR_FRONTIER

⊢ ∀s. interior (frontier s) = interior (closure s) DIFF closure (interior s)

INTERIOR_FINITE_BIGINTER

⊢ ∀s. FINITE s ⇒ (interior (BIGINTER s) = BIGINTER (IMAGE interior s))

INTERIOR_EQ_EMPTY_ALT

⊢ ∀s. (interior s = ∅) ⇔ ∀t. open t ∧ t ≠ ∅ ⇒ t DIFF s ≠ ∅

INTERIOR_EQ_EMPTY

⊢ ∀s. (interior s = ∅) ⇔ ∀t. open t ∧ t ⊆ s ⇒ (t = ∅)

INTERIOR_EQ

⊢ ∀s. (interior s = s) ⇔ open s

INTERIOR_EMPTY

⊢ interior ∅ = ∅

INTERIOR_DIFF

⊢ ∀s t. interior (s DIFF t) = interior s DIFF closure t

INTERIOR_COMPLEMENT

⊢ ∀s. interior (𝕌(:real) DIFF s) = 𝕌(:real) DIFF closure s

INTERIOR_CLOSURE_INTER_OPEN

⊢ ∀s t.
      open s ∧ open t ⇒
      (interior (closure (s ∩ t)) =
       interior (closure s) ∩ interior (closure t))

INTERIOR_CLOSURE_IDEMP

⊢ ∀s. interior (closure (interior (closure s))) = interior (closure s)

INTERIOR_CLOSURE

⊢ ∀s. interior s = 𝕌(:real) DIFF closure (𝕌(:real) DIFF s)

INTERIOR_CLOSED_UNION_EMPTY_INTERIOR

⊢ ∀s t. closed s ∧ (interior t = ∅) ⇒ (interior (s ∪ t) = interior s)

INTERIOR_CLOSED_INTERVAL

⊢ ∀a b. interior (interval [(a,b)]) = interval (a,b)

INTERIOR_CLOSED_EQ_EMPTY_AS_FRONTIER

⊢ ∀s. closed s ∧ (interior s = ∅) ⇔ ∃t. open t ∧ (s = frontier t)

INTERIOR_CBALL

⊢ ∀x e. interior (cball (x,e)) = ball (x,e)

INTERIOR_BIJECTIVE_LINEAR_IMAGE

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ∧ (∀y. ∃x. f x = y) ⇒
      (interior (IMAGE f s) = IMAGE f (interior s))

INTERIOR_BIGINTER_SUBSET

⊢ ∀f. interior (BIGINTER f) ⊆ BIGINTER (IMAGE interior f)

INTERIOR_BALL

⊢ ∀a r. interior (ball (a,r)) = ball (a,r)

INTER_INTERVAL_MIXED_EQ_EMPTY

⊢ ∀a b c d.
      interval (c,d) ≠ ∅ ⇒
      ((interval (a,b) ∩ interval [(c,d)] = ∅) ⇔
       (interval (a,b) ∩ interval (c,d) = ∅))

INTER_INTERVAL

⊢ interval [(a,b)] ∩ interval [(c,d)] = interval [(max a c,min b d)]

INTER_BIGUNION

⊢ (∀s t. BIGUNION s ∩ t = BIGUNION {x ∩ t | x ∈ s}) ∧
  ∀s t. t ∩ BIGUNION s = BIGUNION {t ∩ x | x ∈ s}

INTER_BALLS_EQ_EMPTY

⊢ (∀a b r s.
       (ball (a,r) ∩ ball (b,s) = ∅) ⇔ r ≤ 0 ∨ s ≤ 0 ∨ r + s ≤ dist (a,b)) ∧
  (∀a b r s.
       (ball (a,r) ∩ cball (b,s) = ∅) ⇔ r ≤ 0 ∨ s < 0 ∨ r + s ≤ dist (a,b)) ∧
  (∀a b r s.
       (cball (a,r) ∩ ball (b,s) = ∅) ⇔ r < 0 ∨ s ≤ 0 ∨ r + s ≤ dist (a,b)) ∧
  ∀a b r s.
      (cball (a,r) ∩ cball (b,s) = ∅) ⇔ r < 0 ∨ s < 0 ∨ r + s < dist (a,b)

INJECTIVE_MAP_OPEN_IFF_CLOSED

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      ((∀u.
            open_in (subtopology euclidean s) u ⇒
            open_in (subtopology euclidean t) (IMAGE f u)) ⇔
       ∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u))

INJECTIVE_IMP_ISOMETRIC

⊢ ∀f s.
      closed s ∧ subspace s ∧ linear f ∧ (∀x. x ∈ s ∧ (f x = 0) ⇒ (x = 0)) ⇒
      ∃e. 0 < e ∧ ∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x

INFSUM_UNIQUE

⊢ ∀f l s. (f sums l) s ⇒ (infsum s f = l)

INFSUM_SUB

⊢ ∀x y s.
      summable s x ∧ summable s y ⇒
      (infsum s (λi. x i − y i) = infsum s x − infsum s y)

INFSUM_RESTRICT

⊢ ∀k a. infsum 𝕌(:num) (λn. if n ∈ k then a n else 0) = infsum k a

INFSUM_NEG

⊢ ∀s x. summable s x ⇒ (infsum s (λn. -x n) = -infsum s x)

INFSUM_LINEAR

⊢ ∀f h s. summable s f ∧ linear h ⇒ (infsum s (λn. h (f n)) = h (infsum s f))

INFSUM_EQ

⊢ ∀f g k.
      summable k f ∧ summable k g ∧ (∀x. x ∈ k ⇒ (f x = g x)) ⇒
      (infsum k f = infsum k g)

INFSUM_CMUL

⊢ ∀s x c. summable s x ⇒ (infsum s (λn. c * x n) = c * infsum s x)

INFSUM_ADD

⊢ ∀x y s.
      summable s x ∧ summable s y ⇒
      (infsum s (λi. x i + y i) = infsum s x + infsum s y)

INFSUM_0

⊢ infsum s (λi. 0) = 0

INFINITE_SUPERSET

⊢ ∀s t. INFINITE s ∧ s ⊆ t ⇒ INFINITE t

INFINITE_OPEN_IN

⊢ ∀u s.
      open_in (subtopology euclidean u) s ∧ (∃x. x ∈ s ∧ x limit_point_of u) ⇒
      INFINITE s

INFINITE_FROM

⊢ ∀n. INFINITE (from n)

INF_INSERT

⊢ ∀x s. bounded s ⇒ (inf (x INSERT s) = if s = ∅ then x else min x (inf s))

INF_FINITE_LEMMA

⊢ ∀s. FINITE s ∧ s ≠ ∅ ⇒ ∃b. b ∈ s ∧ ∀x. x ∈ s ⇒ b ≤ x

INF_FINITE

⊢ ∀s. FINITE s ∧ s ≠ ∅ ⇒ inf s ∈ s ∧ ∀x. x ∈ s ⇒ inf s ≤ x

INDEPENDENT_STDBASIS

⊢ independent {i | 1 ≤ i ∧ i ≤ 1}

INDEPENDENT_SPAN_BOUND

⊢ ∀s t. FINITE t ∧ independent s ∧ s ⊆ span t ⇒ FINITE s ∧ CARD s ≤ CARD t

INDEPENDENT_SING

⊢ ∀x. independent {x} ⇔ x ≠ 0

INDEPENDENT_NONZERO

⊢ ∀s. independent s ⇒ 0 ∉ s

INDEPENDENT_MONO

⊢ ∀s t. independent t ∧ s ⊆ t ⇒ independent s

INDEPENDENT_INSERT

⊢ ∀a s.
      independent (a INSERT s) ⇔
      if a ∈ s then independent s else independent s ∧ a ∉ span s

INDEPENDENT_INJECTIVE_IMAGE_GEN

⊢ ∀f s.
      independent s ∧ linear f ∧
      (∀x y. x ∈ span s ∧ y ∈ span s ∧ (f x = f y) ⇒ (x = y)) ⇒
      independent (IMAGE f s)

INDEPENDENT_INJECTIVE_IMAGE

⊢ ∀f s.
      independent s ∧ linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      independent (IMAGE f s)

INDEPENDENT_EMPTY

⊢ independent ∅

INDEPENDENT_CARD_LE_DIM

⊢ ∀v b. b ⊆ v ∧ independent b ⇒ FINITE b ∧ CARD b ≤ dim v

INDEPENDENT_BOUND

⊢ ∀s. independent s ⇒ FINITE s ∧ CARD s ≤ 1

IN_SPHERE_0

⊢ ∀x e. x ∈ sphere (0,e) ⇔ (abs x = e)

IN_SPHERE

⊢ ∀x y e. y ∈ sphere (x,e) ⇔ (dist (x,y) = e)

IN_SPAN_INSERT

⊢ ∀a b s. a ∈ span (b INSERT s) ∧ a ∉ span s ⇒ b ∈ span (a INSERT s)

IN_SPAN_DELETE

⊢ ∀a b s. a ∈ span s ∧ a ∉ span (s DELETE b) ⇒ b ∈ span (a INSERT s DELETE b)

IN_SEGMENT_COMPONENT

⊢ ∀a b x i. x ∈ segment [(a,b)] ⇒ min a b ≤ x ∧ x ≤ max a b

IN_SEGMENT

⊢ ∀a b x.
      (x ∈ segment [(a,b)] ⇔ ∃u. 0 ≤ u ∧ u ≤ 1 ∧ (x = (1 − u) * a + u * b)) ∧
      (x ∈ segment (a,b) ⇔
       a ≠ b ∧ ∃u. 0 < u ∧ u < 1 ∧ (x = (1 − u) * a + u * b))

IN_OPEN_SEGMENT_ALT

⊢ ∀a b x. x ∈ segment (a,b) ⇔ x ∈ segment [(a,b)] ∧ x ≠ a ∧ x ≠ b ∧ a ≠ b

IN_OPEN_SEGMENT

⊢ ∀a b x. x ∈ segment (a,b) ⇔ x ∈ segment [(a,b)] ∧ x ≠ a ∧ x ≠ b

IN_INTERVAL_REFLECT

⊢ (∀a b x. -x ∈ interval [(-b,-a)] ⇔ x ∈ interval [(a,b)]) ∧
  ∀a b x. -x ∈ interval (-b,-a) ⇔ x ∈ interval (a,b)

IN_INTERVAL

⊢ (x ∈ interval (a,b) ⇔ a < x ∧ x < b) ∧
  (x ∈ interval [(a,b)] ⇔ a ≤ x ∧ x ≤ b)

IN_INTERIOR_LINEAR_IMAGE

⊢ ∀f g s x.
      linear f ∧ linear g ∧ (f ∘ g = I) ∧ x ∈ interior s ⇒
      f x ∈ interior (IMAGE f s)

IN_INTERIOR_CBALL

⊢ ∀x s. x ∈ interior s ⇔ ∃e. 0 < e ∧ cball (x,e) ⊆ s

IN_INTERIOR

⊢ ∀x s. x ∈ interior s ⇔ ∃e. 0 < e ∧ ball (x,e) ⊆ s

IN_DIRECTION

⊢ ∀a v x y.
      netord (a in_direction v) x y ⇔
      0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a) ∧ ∃c. 0 ≤ c ∧ (x − a = c * v)

IN_COMPONENTS_SUBSET

⊢ ∀s c. c ∈ components s ⇒ c ⊆ s

IN_COMPONENTS_SELF

⊢ ∀s. s ∈ components s ⇔ connected s ∧ s ≠ ∅

IN_COMPONENTS_NONEMPTY

⊢ ∀s c. c ∈ components s ⇒ c ≠ ∅

IN_COMPONENTS_MAXIMAL

⊢ ∀s c.
      c ∈ components s ⇔
      c ≠ ∅ ∧ c ⊆ s ∧ connected c ∧
      ∀c'. c' ≠ ∅ ∧ c ⊆ c' ∧ c' ⊆ s ∧ connected c' ⇒ (c' = c)

IN_COMPONENTS_CONNECTED

⊢ ∀s c. c ∈ components s ⇒ connected c

IN_COMPONENTS_BIGUNION_COMPLEMENT

⊢ ∀s c. c ∈ components s ⇒ (s DIFF c = BIGUNION (components s DELETE c))

IN_COMPONENTS

⊢ ∀u s. s ∈ components u ⇔ ∃x. x ∈ u ∧ (s = connected_component u x)

IN_CLOSURE_DELETE

⊢ ∀s x. x ∈ closure (s DELETE x) ⇔ x limit_point_of s

IN_CBALL_0

⊢ ∀x e. x ∈ cball (0,e) ⇔ abs x ≤ e

IN_CBALL

⊢ ∀x y e. y ∈ cball (x,e) ⇔ dist (x,y) ≤ e

IN_BALL_0

⊢ ∀x e. x ∈ ball (0,e) ⇔ abs x < e

IN_BALL

⊢ ∀x y e. y ∈ ball (x,e) ⇔ dist (x,y) < e

IMAGE_TWIZZLE_INTERVAL

⊢ ∀p a b. IMAGE (λx. x) (interval [(a,b)]) = interval [(a,b)]

IMAGE_STRETCH_INTERVAL

⊢ ∀a b m.
      IMAGE (λx. @f. f = m 1 * x) (interval [(a,b)]) =
      if interval [(a,b)] = ∅ then ∅
      else
        interval
          [((@f. f = min (m 1 * a) (m 1 * b)),@f. f = max (m 1 * a) (m 1 * b))]

IMAGE_SING

⊢ ∀f a. IMAGE f {a} = {f a}

IMAGE_CLOSURE_SUBSET

⊢ ∀f s t.
      f continuous_on closure s ∧ closed t ∧ IMAGE f s ⊆ t ⇒
      IMAGE f (closure s) ⊆ t

IMAGE_AFFINITY_INTERVAL

⊢ ∀a b m c.
      IMAGE (λx. m * x + c) (interval [(a,b)]) =
      if interval [(a,b)] = ∅ then ∅
      else if 0 ≤ m then interval [(m * a + c,m * b + c)]
      else interval [(m * b + c,m * a + c)]

HOMEOMORPHISM_SYM

⊢ ∀f g s t. homeomorphism (s,t) (f,g) ⇔ homeomorphism (t,s) (g,f)

HOMEOMORPHISM_OF_SUBSETS

⊢ ∀f g s t s' t'.
      homeomorphism (s,t) (f,g) ∧ s' ⊆ s ∧ t' ⊆ t ∧ (IMAGE f s' = t') ⇒
      homeomorphism (s',t') (f,g)

HOMEOMORPHISM_LOCALLY

⊢ ∀P Q f g.
      (∀s t. homeomorphism (s,t) (f,g) ⇒ (P s ⇔ Q t)) ⇒
      ∀s t. homeomorphism (s,t) (f,g) ⇒ (locally P s ⇔ locally Q t)

HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      ((∃g. homeomorphism (s,t) (f,g)) ⇔
       ∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t) (IMAGE f u))

HOMEOMORPHISM_INJECTIVE_OPEN_MAP

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ∧
      (∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t) (IMAGE f u)) ⇒
      ∃g. homeomorphism (s,t) (f,g)

HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      ((∃g. homeomorphism (s,t) (f,g)) ⇔
       ∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u))

HOMEOMORPHISM_INJECTIVE_CLOSED_MAP

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ∧
      (∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u)) ⇒
      ∃g. homeomorphism (s,t) (f,g)

HOMEOMORPHISM_IMP_QUOTIENT_MAP

⊢ ∀f g s t.
      homeomorphism (s,t) (f,g) ⇒
      ∀u.
          u ⊆ t ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
           open_in (subtopology euclidean t) u)

HOMEOMORPHISM_IMP_OPEN_MAP

⊢ ∀f g s t u.
      homeomorphism (s,t) (f,g) ∧ open_in (subtopology euclidean s) u ⇒
      open_in (subtopology euclidean t) (IMAGE f u)

HOMEOMORPHISM_IMP_CLOSED_MAP

⊢ ∀f g s t u.
      homeomorphism (s,t) (f,g) ∧ closed_in (subtopology euclidean s) u ⇒
      closed_in (subtopology euclidean t) (IMAGE f u)

HOMEOMORPHISM_ID

⊢ ∀s. homeomorphism (s,s) ((λx. x),(λx. x))

HOMEOMORPHISM_FROM_COMPOSITION_SURJECTIVE

⊢ ∀f g s t u.
      f continuous_on s ∧ (IMAGE f s = t) ∧ g continuous_on t ∧
      IMAGE g t ⊆ u ∧ (∃h. homeomorphism (s,u) (g ∘ f,h)) ⇒
      (∃f'. homeomorphism (s,t) (f,f')) ∧ ∃g'. homeomorphism (t,u) (g,g')

HOMEOMORPHISM_FROM_COMPOSITION_INJECTIVE

⊢ ∀f g s t u.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧ IMAGE g t ⊆ u ∧
      (∀x y. x ∈ t ∧ y ∈ t ∧ (g x = g y) ⇒ (x = y)) ∧
      (∃h. homeomorphism (s,u) (g ∘ f,h)) ⇒
      (∃f'. homeomorphism (s,t) (f,f')) ∧ ∃g'. homeomorphism (t,u) (g,g')

HOMEOMORPHISM_COMPOSE

⊢ ∀f g h k s t u.
      homeomorphism (s,t) (f,g) ∧ homeomorphism (t,u) (h,k) ⇒
      homeomorphism (s,u) (h ∘ f,g ∘ k)

HOMEOMORPHISM_COMPACT

⊢ ∀s f t.
      compact s ∧ f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      ∃g. homeomorphism (s,t) (f,g)

HOMEOMORPHISM

⊢ ∀s t f g.
      homeomorphism (s,t) (f,g) ⇔
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧ IMAGE g t ⊆ s ∧
      (∀x. x ∈ s ⇒ (g (f x) = x)) ∧ ∀y. y ∈ t ⇒ (f (g y) = y)

HOMEOMORPHIC_TRANSLATION_SELF

⊢ ∀a s. IMAGE (λx. a + x) s homeomorphic s

HOMEOMORPHIC_TRANSLATION_RIGHT_EQ

⊢ ∀a s t. s homeomorphic IMAGE (λx. a + x) t ⇔ s homeomorphic t

HOMEOMORPHIC_TRANSLATION_LEFT_EQ

⊢ ∀a s t. IMAGE (λx. a + x) s homeomorphic t ⇔ s homeomorphic t

HOMEOMORPHIC_TRANSLATION

⊢ ∀s a. s homeomorphic IMAGE (λx. a + x) s

HOMEOMORPHIC_TRANS

⊢ ∀s t u. s homeomorphic t ∧ t homeomorphic u ⇒ s homeomorphic u

HOMEOMORPHIC_SYM

⊢ ∀s t. s homeomorphic t ⇔ t homeomorphic s

HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE

⊢ ∀a b c. a ≠ 0 ⇒ {x | x = c} homeomorphic {x | a * x = b}

HOMEOMORPHIC_SPHERE

⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ sphere (a,d) homeomorphic sphere (b,e)

HOMEOMORPHIC_SING

⊢ ∀a b. {a} homeomorphic {b}

HOMEOMORPHIC_SCALING_RIGHT

⊢ ∀c. 0 < c ⇒ ∀s t. s homeomorphic IMAGE (λx. c * x) t ⇔ s homeomorphic t

HOMEOMORPHIC_SCALING_LEFT

⊢ ∀c. 0 < c ⇒ ∀s t. IMAGE (λx. c * x) s homeomorphic t ⇔ s homeomorphic t

HOMEOMORPHIC_SCALING

⊢ ∀s c. c ≠ 0 ⇒ s homeomorphic IMAGE (λx. c * x) s

HOMEOMORPHIC_REFL

⊢ ∀s. s homeomorphic s

HOMEOMORPHIC_OPEN_INTERVALS

⊢ ∀a b c d. a < b ∧ c < d ⇒ interval (a,b) homeomorphic interval (c,d)

HOMEOMORPHIC_OPEN_INTERVAL_UNIV

⊢ ∀a b. a < b ⇒ interval (a,b) homeomorphic 𝕌(:real)

HOMEOMORPHIC_ONE_POINT_COMPACTIFICATIONS

⊢ ∀s t a b.
      compact s ∧ compact t ∧ a ∈ s ∧ b ∈ t ∧
      s DELETE a homeomorphic t DELETE b ⇒
      s homeomorphic t

HOMEOMORPHIC_MINIMAL

⊢ ∀s t.
      s homeomorphic t ⇔
      ∃f g.
          (∀x. x ∈ s ⇒ f x ∈ t ∧ (g (f x) = x)) ∧
          (∀y. y ∈ t ⇒ g y ∈ s ∧ (f (g y) = y)) ∧ f continuous_on s ∧
          g continuous_on t

HOMEOMORPHIC_LOCALLY

⊢ ∀P Q.
      (∀s t. s homeomorphic t ⇒ (P s ⇔ Q t)) ⇒
      ∀s t. s homeomorphic t ⇒ (locally P s ⇔ locally Q t)

HOMEOMORPHIC_LOCAL_COMPACTNESS

⊢ ∀s t. s homeomorphic t ⇒ (locally compact s ⇔ locally compact t)

HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF

⊢ ∀f s. linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒ IMAGE f s homeomorphic s

HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ

⊢ ∀f s t.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (s homeomorphic IMAGE f t ⇔ s homeomorphic t)

HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ

⊢ ∀f s t.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (IMAGE f s homeomorphic t ⇔ s homeomorphic t)

HOMEOMORPHIC_IMP_CARD_EQ

⊢ ∀s t. s homeomorphic t ⇒ s ≈ t

HOMEOMORPHIC_HYPERPLANES

⊢ ∀a b c d. a ≠ 0 ∧ c ≠ 0 ⇒ {x | a * x = b} homeomorphic {x | c * x = d}

HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE

⊢ ∀a b c. a ≠ 0 ⇒ {x | a * x = b} homeomorphic {x | x = c}

HOMEOMORPHIC_FINITENESS

⊢ ∀s t. s homeomorphic t ⇒ (FINITE s ⇔ FINITE t)

HOMEOMORPHIC_FINITE_STRONG

⊢ ∀s t.
      FINITE s ∨ FINITE t ⇒
      (s homeomorphic t ⇔ FINITE s ∧ FINITE t ∧ (CARD s = CARD t))

HOMEOMORPHIC_FINITE

⊢ ∀s t. FINITE s ∧ FINITE t ⇒ (s homeomorphic t ⇔ (CARD s = CARD t))

HOMEOMORPHIC_EMPTY

⊢ (∀s. s homeomorphic ∅ ⇔ (s = ∅)) ∧ ∀s. ∅ homeomorphic s ⇔ (s = ∅)

HOMEOMORPHIC_CONNECTEDNESS

⊢ ∀s t. s homeomorphic t ⇒ (connected s ⇔ connected t)

HOMEOMORPHIC_COMPACTNESS

⊢ ∀s t. s homeomorphic t ⇒ (compact s ⇔ compact t)

HOMEOMORPHIC_COMPACT

⊢ ∀s f t.
      compact s ∧ f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      s homeomorphic t

HOMEOMORPHIC_CBALL

⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ cball (a,d) homeomorphic cball (b,e)

HOMEOMORPHIC_BALLS_CBALL_SPHERE

⊢ (∀a b d e. 0 < d ∧ 0 < e ⇒ ball (a,d) homeomorphic ball (b,e)) ∧
  (∀a b d e. 0 < d ∧ 0 < e ⇒ cball (a,d) homeomorphic cball (b,e)) ∧
  ∀a b d e. 0 < d ∧ 0 < e ⇒ sphere (a,d) homeomorphic sphere (b,e)

HOMEOMORPHIC_BALLS

⊢ ∀a b d e. 0 < d ∧ 0 < e ⇒ ball (a,d) homeomorphic ball (b,e)

HOMEOMORPHIC_AFFINITY

⊢ ∀s a c. c ≠ 0 ⇒ s homeomorphic IMAGE (λx. a + c * x) s

HEINE_BOREL_LEMMA

⊢ ∀s.
      compact s ⇒
      ∀t.
          s ⊆ BIGUNION t ∧ (∀b. b ∈ t ⇒ open b) ⇒
          ∃e. 0 < e ∧ ∀x. x ∈ s ⇒ ∃b. b ∈ t ∧ ball (x,e) ⊆ b

HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS

⊢ ∀s.
      (∀f.
           (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
           ∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f') ⇒
      ∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t

HAUSDIST_UNION_LE

⊢ ∀s t u.
      bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ∧ u ≠ ∅ ⇒
      hausdist (s ∪ t,s ∪ u) ≤ hausdist (t,u)

HAUSDIST_TRIANGLE

⊢ ∀s t u.
      bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ⇒
      hausdist (s,u) ≤ hausdist (s,t) + hausdist (t,u)

HAUSDIST_TRANSLATION

⊢ ∀a s t. hausdist (IMAGE (λx. a + x) s,IMAGE (λx. a + x) t) = hausdist (s,t)

HAUSDIST_TRANS

⊢ ∀s t u.
      bounded s ∧ bounded t ∧ bounded u ∧ t ≠ ∅ ⇒
      hausdist (s,u) ≤ hausdist (s,t) + hausdist (t,u)

HAUSDIST_SYM

⊢ ∀s t. hausdist (s,t) = hausdist (t,s)

HAUSDIST_SINGS

⊢ ∀x y. hausdist ({x},{y}) = dist (x,y)

HAUSDIST_SETDIST_TRIANGLE

⊢ ∀s t u.
      t ≠ ∅ ∧ bounded s ∧ bounded t ⇒
      setdist (s,u) ≤ hausdist (s,t) + setdist (t,u)

HAUSDIST_REFL

⊢ ∀s. hausdist (s,s) = 0

HAUSDIST_POS_LE

⊢ ∀s t. 0 ≤ hausdist (s,t)

HAUSDIST_NONTRIVIAL_ALT

⊢ ∀s t.
      bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      (hausdist (s,t) =
       max (sup {setdist ({x},t) | x ∈ s}) (sup {setdist ({y},s) | y ∈ t}))

HAUSDIST_NONTRIVIAL

⊢ ∀s t.
      bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      (hausdist (s,t) =
       sup ({setdist ({x},t) | x ∈ s} ∪ {setdist ({y},s) | y ∈ t}))

HAUSDIST_LINEAR_IMAGE

⊢ ∀f s t.
      linear f ∧ (∀x. abs (f x) = abs x) ⇒
      (hausdist (IMAGE f s,IMAGE f t) = hausdist (s,t))

HAUSDIST_INSERT_LE

⊢ ∀s t a.
      bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      hausdist (a INSERT s,a INSERT t) ≤ hausdist (s,t)

HAUSDIST_EQ_0

⊢ ∀s t.
      bounded s ∧ bounded t ⇒
      ((hausdist (s,t) = 0) ⇔ (s = ∅) ∨ (t = ∅) ∨ (closure s = closure t))

HAUSDIST_EQ

⊢ ∀s t s' t'.
      (∀b.
           (∀x. x ∈ s ⇒ setdist ({x},t) ≤ b) ∧
           (∀y. y ∈ t ⇒ setdist ({y},s) ≤ b) ⇔
           (∀x. x ∈ s' ⇒ setdist ({x},t') ≤ b) ∧
           ∀y. y ∈ t' ⇒ setdist ({y},s') ≤ b) ⇒
      (hausdist (s,t) = hausdist (s',t'))

HAUSDIST_EMPTY

⊢ (∀t. hausdist (∅,t) = 0) ∧ ∀s. hausdist (s,∅) = 0

HAUSDIST_COMPACT_SUMS

⊢ ∀s t.
      bounded s ∧ compact t ∧ t ≠ ∅ ⇒
      s ⊆ {y + z | y ∈ t ∧ z ∈ cball (0,hausdist (s,t))}

HAUSDIST_COMPACT_NONTRIVIAL

⊢ ∀s t.
      compact s ∧ compact t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      (hausdist (s,t) =
       inf
         {e |
          0 ≤ e ∧ s ⊆ {x + y | x ∈ t ∧ abs y ≤ e} ∧
          t ⊆ {x + y | x ∈ s ∧ abs y ≤ e}})

HAUSDIST_COMPACT_EXISTS

⊢ ∀s t.
      bounded s ∧ compact t ∧ t ≠ ∅ ⇒
      ∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ dist (x,y) ≤ hausdist (s,t)

HAUSDIST_CLOSURE

⊢ (∀s t. hausdist (closure s,t) = hausdist (s,t)) ∧
  ∀s t. hausdist (s,closure t) = hausdist (s,t)

HAUSDIST_ALT

⊢ ∀s t.
      bounded s ∧ bounded t ∧ s ≠ ∅ ∧ t ≠ ∅ ⇒
      (hausdist (s,t) =
       sup {abs (setdist ({x},s) − setdist ({x},t)) | x ∈ 𝕌(:real)})

HAS_SIZE_STDBASIS

⊢ {i | 1 ≤ i ∧ i ≤ 1} HAS_SIZE 1

GREATER_EQ_REFL

⊢ ∀m. m ≥ m

GDELTA_COMPLEMENT

⊢ ∀s. gdelta (𝕌(:real) DIFF s) ⇔ fsigma s

FUNCTION_FACTORS_LEFT_GEN

⊢ ∀P f g.
      (∀x y. P x ∧ P y ∧ (g x = g y) ⇒ (f x = f y)) ⇔
      ∃h. ∀x. P x ⇒ (f x = h (g x))

FUN_IN_IMAGE

⊢ ∀f s x. x ∈ s ⇒ f x ∈ IMAGE f s

FSIGMA_COMPLEMENT

⊢ ∀s. fsigma (𝕌(:real) DIFF s) ⇔ gdelta s

FRONTIER_UNIV

⊢ frontier 𝕌(:real) = ∅

FRONTIER_UNION_SUBSET

⊢ ∀s t. frontier (s ∪ t) ⊆ frontier s ∪ frontier t

FRONTIER_UNION

⊢ ∀s t.
      (closure s ∩ closure t = ∅) ⇒
      (frontier (s ∪ t) = frontier s ∪ frontier t)

FRONTIER_SUBSET_EQ

⊢ ∀s. frontier s ⊆ s ⇔ closed s

FRONTIER_SUBSET_COMPACT

⊢ ∀s. compact s ⇒ frontier s ⊆ s

FRONTIER_SUBSET_CLOSED

⊢ ∀s. closed s ⇒ frontier s ⊆ s

FRONTIER_STRADDLE

⊢ ∀a s.
      a ∈ frontier s ⇔
      ∀e. 0 < e ⇒ (∃x. x ∈ s ∧ dist (a,x) < e) ∧ ∃x. x ∉ s ∧ dist (a,x) < e

FRONTIER_SING

⊢ ∀a. frontier {a} = {a}

FRONTIER_OPEN_INTERVAL

⊢ ∀a b.
      frontier (interval (a,b)) =
      if interval (a,b) = ∅ then ∅ else interval [(a,b)] DIFF interval (a,b)

FRONTIER_INTERIORS

⊢ ∀s. frontier s = 𝕌(:real) DIFF interior s DIFF interior (𝕌(:real) DIFF s)

FRONTIER_INTERIOR_SUBSET

⊢ ∀s. frontier (interior s) ⊆ frontier s

FRONTIER_INTER_SUBSET_INTER

⊢ ∀s t. frontier (s ∩ t) ⊆ closure s ∩ frontier t ∪ frontier s ∩ closure t

FRONTIER_INTER_SUBSET

⊢ ∀s t. frontier (s ∩ t) ⊆ frontier s ∪ frontier t

FRONTIER_HALFSPACE_LT

⊢ ∀a b. ¬((a = 0) ∧ (b = 0)) ⇒ (frontier {x | a * x < b} = {x | a * x = b})

FRONTIER_HALFSPACE_LE

⊢ ∀a b. ¬((a = 0) ∧ (b = 0)) ⇒ (frontier {x | a * x ≤ b} = {x | a * x = b})

FRONTIER_HALFSPACE_GT

⊢ ∀a b. ¬((a = 0) ∧ (b = 0)) ⇒ (frontier {x | a * x > b} = {x | a * x = b})

FRONTIER_HALFSPACE_GE

⊢ ∀a b. ¬((a = 0) ∧ (b = 0)) ⇒ (frontier {x | a * x ≥ b} = {x | a * x = b})

FRONTIER_FRONTIER_SUBSET

⊢ ∀s. frontier (frontier s) ⊆ frontier s

FRONTIER_FRONTIER_FRONTIER

⊢ ∀s. frontier (frontier (frontier s)) = frontier (frontier s)

FRONTIER_FRONTIER

⊢ ∀s. open s ∨ closed s ⇒ (frontier (frontier s) = frontier s)

FRONTIER_EMPTY

⊢ frontier ∅ = ∅

FRONTIER_DISJOINT_EQ

⊢ ∀s. (frontier s ∩ s = ∅) ⇔ open s

FRONTIER_COMPLEMENT

⊢ ∀s. frontier (𝕌(:real) DIFF s) = frontier s

FRONTIER_CLOSURES

⊢ ∀s. frontier s = closure s ∩ closure (𝕌(:real) DIFF s)

FRONTIER_CLOSURE_SUBSET

⊢ ∀s. frontier (closure s) ⊆ frontier s

FRONTIER_CLOSED_INTERVAL

⊢ ∀a b. frontier (interval [(a,b)]) = interval [(a,b)] DIFF interval (a,b)

FRONTIER_CLOSED

⊢ ∀s. closed (frontier s)

FRONTIER_CBALL

⊢ ∀a e. frontier (cball (a,e)) = sphere (a,e)

FRONTIER_BALL

⊢ ∀a e. 0 < e ⇒ (frontier (ball (a,e)) = sphere (a,e))

FROM_INTER_NUMSEG_MAX

⊢ ∀m n p. from p ∩ (m .. n) = MAX p m .. n

FROM_INTER_NUMSEG_GEN

⊢ ∀k m n. from k ∩ (m .. n) = if m < k then k .. n else m .. n

FROM_INTER_NUMSEG

⊢ ∀k n. from k ∩ (0 .. n) = k .. n

FORALL_SUC

⊢ (∀n. n ≠ 0 ⇒ P n) ⇔ ∀n. P (SUC n)

FORALL_POS_MONO_1

⊢ ∀P. (∀d e. d < e ∧ P d ⇒ P e) ∧ (∀n. P (&n + 1)⁻¹) ⇒ ∀e. 0 < e ⇒ P e

FORALL_POS_MONO

⊢ ∀P. (∀d e. d < e ∧ P d ⇒ P e) ∧ (∀n. n ≠ 0 ⇒ P (&n)⁻¹) ⇒ ∀e. 0 < e ⇒ P e

FORALL_IN_GSPEC

⊢ (∀P f. (∀z. z ∈ {f x | P x} ⇒ Q z) ⇔ ∀x. P x ⇒ Q (f x)) ∧
  (∀P f. (∀z. z ∈ {f x y | P x y} ⇒ Q z) ⇔ ∀x y. P x y ⇒ Q (f x y)) ∧
  ∀P f. (∀z. z ∈ {f w x y | P w x y} ⇒ Q z) ⇔ ∀w x y. P w x y ⇒ Q (f w x y)

FORALL_IN_CLOSURE_EQ

⊢ ∀f s t.
      closed t ∧ f continuous_on closure s ⇒
      ((∀x. x ∈ closure s ⇒ f x ∈ t) ⇔ ∀x. x ∈ s ⇒ f x ∈ t)

FORALL_IN_CLOSURE

⊢ ∀f s t.
      closed t ∧ f continuous_on closure s ∧ (∀x. x ∈ s ⇒ f x ∈ t) ⇒
      ∀x. x ∈ closure s ⇒ f x ∈ t

FORALL_FINITE_SUBSET_IMAGE

⊢ ∀P f s.
      (∀t. FINITE t ∧ t ⊆ IMAGE f s ⇒ P t) ⇔
      ∀t. FINITE t ∧ t ⊆ s ⇒ P (IMAGE f t)

FORALL_EVENTUALLY

⊢ ∀net p s.
      FINITE s ∧ s ≠ ∅ ⇒
      ((∀a. a ∈ s ⇒ eventually (p a) net) ⇔
       eventually (λx. ∀a. a ∈ s ⇒ p a x) net)

FINITE_SUBSET_IMAGE

⊢ ∀f s t.
      FINITE t ∧ t ⊆ IMAGE f s ⇔ ∃s'. FINITE s' ∧ s' ⊆ s ∧ (t = IMAGE f s')

FINITE_SPHERE

⊢ ∀a r. FINITE (sphere (a,r))

FINITE_SET_AVOID

⊢ ∀a s. FINITE s ⇒ ∃d. 0 < d ∧ ∀x. x ∈ s ∧ x ≠ a ⇒ d ≤ dist (a,x)

FINITE_POWERSET

⊢ ∀s. FINITE s ⇒ FINITE {t | t ⊆ s}

FINITE_INTERVAL

⊢ (∀a b. FINITE (interval [(a,b)]) ⇔ b ≤ a) ∧
  ∀a b. FINITE (interval (a,b)) ⇔ b ≤ a

FINITE_INTER_NUMSEG

⊢ ∀s m n. FINITE (s ∩ (m .. n))

FINITE_IMP_NOT_OPEN

⊢ ∀s. FINITE s ∧ s ≠ ∅ ⇒ ¬open s

FINITE_IMP_COMPACT

⊢ ∀s. FINITE s ⇒ compact s

FINITE_IMP_CLOSED_IN

⊢ ∀s t. FINITE s ∧ s ⊆ t ⇒ closed_in (subtopology euclidean t) s

FINITE_IMP_CLOSED

⊢ ∀s. FINITE s ⇒ closed s

FINITE_IMP_BOUNDED

⊢ ∀s. FINITE s ⇒ bounded s

FINITE_CBALL

⊢ ∀a r. FINITE (cball (a,r)) ⇔ r ≤ 0

FINITE_BALL

⊢ ∀a r. FINITE (ball (a,r)) ⇔ r ≤ 0

EXTENSION_FROM_CLOPEN

⊢ ∀f s t u.
      open_in (subtopology euclidean s) t ∧
      closed_in (subtopology euclidean s) t ∧ f continuous_on t ∧
      IMAGE f t ⊆ u ∧ ((u = ∅) ⇒ (s = ∅)) ⇒
      ∃g. g continuous_on s ∧ IMAGE g s ⊆ u ∧ ∀x. x ∈ t ⇒ (g x = f x)

EXISTS_IN_INSERT

⊢ ∀P a s. (∃x. x ∈ a INSERT s ∧ P x) ⇔ P a ∨ ∃x. x ∈ s ∧ P x

EXISTS_IN_GSPEC

⊢ (∀P f. (∃z. z ∈ {f x | P x} ∧ Q z) ⇔ ∃x. P x ∧ Q (f x)) ∧
  (∀P f. (∃z. z ∈ {f x y | P x y} ∧ Q z) ⇔ ∃x y. P x y ∧ Q (f x y)) ∧
  ∀P f. (∃z. z ∈ {f w x y | P w x y} ∧ Q z) ⇔ ∃w x y. P w x y ∧ Q (f w x y)

EXISTS_FINITE_SUBSET_IMAGE

⊢ ∀P f s.
      (∃t. FINITE t ∧ t ⊆ IMAGE f s ∧ P t) ⇔
      ∃t. FINITE t ∧ t ⊆ s ∧ P (IMAGE f t)

EXISTS_DIFF

⊢ (∃s. P (𝕌(:α) DIFF s)) ⇔ ∃s. P s

EXISTS_COMPONENT_SUPERSET

⊢ ∀s t. t ⊆ s ∧ s ≠ ∅ ∧ connected t ⇒ ∃c. c ∈ components s ∧ t ⊆ c

EXCHANGE_LEMMA

⊢ ∀s t.
      FINITE t ∧ independent s ∧ s ⊆ span t ⇒
      ∃t'. t' HAS_SIZE CARD t ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'

EVENTUALLY_WITHIN_LE

⊢ ∀s a p.
      eventually p (at a within s) ⇔
      ∃d. 0 < d ∧ ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) ≤ d ⇒ p x

EVENTUALLY_WITHIN_INTERIOR

⊢ ∀p s x.
      x ∈ interior s ⇒ (eventually p (at x within s) ⇔ eventually p (at x))

EVENTUALLY_WITHIN

⊢ ∀s a p.
      eventually p (at a within s) ⇔
      ∃d. 0 < d ∧ ∀x. x ∈ s ∧ 0 < dist (x,a) ∧ dist (x,a) < d ⇒ p x

EVENTUALLY_TRUE

⊢ ∀net. eventually (λx. T) net ⇔ T

EVENTUALLY_SEQUENTIALLY

⊢ ∀p. eventually p sequentially ⇔ ∃N. ∀n. N ≤ n ⇒ p n

EVENTUALLY_MP

⊢ ∀net p q.
      eventually (λx. p x ⇒ q x) net ∧ eventually p net ⇒ eventually q net

EVENTUALLY_MONO

⊢ ∀net p q. (∀x. p x ⇒ q x) ∧ eventually p net ⇒ eventually q net

EVENTUALLY_HAPPENS

⊢ ∀net p. eventually p net ⇒ trivial_limit net ∨ ∃x. p x

EVENTUALLY_FORALL

⊢ ∀net p s.
      FINITE s ∧ s ≠ ∅ ⇒
      (eventually (λx. ∀a. a ∈ s ⇒ p a x) net ⇔
       ∀a. a ∈ s ⇒ eventually (p a) net)

EVENTUALLY_FALSE

⊢ ∀net. eventually (λx. F) net ⇔ trivial_limit net

EVENTUALLY_AT_POSINFINITY

⊢ ∀p. eventually p at_posinfinity ⇔ ∃b. ∀x. x ≥ b ⇒ p x

EVENTUALLY_AT_NEGINFINITY

⊢ ∀p. eventually p at_neginfinity ⇔ ∃b. ∀x. x ≤ b ⇒ p x

EVENTUALLY_AT_INFINITY_POS

⊢ ∀p. eventually p at_infinity ⇔ ∃b. 0 < b ∧ ∀x. abs x ≥ b ⇒ p x

EVENTUALLY_AT_INFINITY

⊢ ∀p. eventually p at_infinity ⇔ ∃b. ∀x. abs x ≥ b ⇒ p x

EVENTUALLY_AT

⊢ ∀a p.
      eventually p (at a) ⇔
      ∃d. 0 < d ∧ ∀x. 0 < dist (x,a) ∧ dist (x,a) < d ⇒ p x

EVENTUALLY_AND

⊢ ∀net p q.
      eventually (λx. p x ∧ q x) net ⇔ eventually p net ∧ eventually q net

EQ_INTERVAL

⊢ (∀a b c d.
       (interval [(a,b)] = interval [(c,d)]) ⇔
       (interval [(a,b)] = ∅) ∧ (interval [(c,d)] = ∅) ∨ (a = c) ∧ (b = d)) ∧
  (∀a b c d.
       (interval [(a,b)] = interval (c,d)) ⇔
       (interval [(a,b)] = ∅) ∧ (interval (c,d) = ∅)) ∧
  (∀a b c d.
       (interval (a,b) = interval [(c,d)]) ⇔
       (interval (a,b) = ∅) ∧ (interval [(c,d)] = ∅)) ∧
  ∀a b c d.
      (interval (a,b) = interval (c,d)) ⇔
      (interval (a,b) = ∅) ∧ (interval (c,d) = ∅) ∨ (a = c) ∧ (b = d)

EQ_BALLS

⊢ (∀a a' r r'.
       (ball (a,r) = ball (a',r')) ⇔ (a = a') ∧ (r = r') ∨ r ≤ 0 ∧ r' ≤ 0) ∧
  (∀a a' r r'. (ball (a,r) = cball (a',r')) ⇔ r ≤ 0 ∧ r' < 0) ∧
  (∀a a' r r'. (cball (a,r) = ball (a',r')) ⇔ r < 0 ∧ r' ≤ 0) ∧
  ∀a a' r r'.
      (cball (a,r) = cball (a',r')) ⇔ (a = a') ∧ (r = r') ∨ r < 0 ∧ r' < 0

ENDS_NOT_IN_SEGMENT

⊢ ∀a b. a ∉ segment (a,b) ∧ b ∉ segment (a,b)

ENDS_IN_UNIT_INTERVAL

⊢ 0 ∈ interval [(0,1)] ∧ 1 ∈ interval [(0,1)] ∧ 0 ∉ interval (0,1) ∧
  1 ∉ interval (0,1)

ENDS_IN_SEGMENT

⊢ ∀a b. a ∈ segment [(a,b)] ∧ b ∈ segment [(a,b)]

ENDS_IN_INTERVAL

⊢ (∀a b. a ∈ interval [(a,b)] ⇔ interval [(a,b)] ≠ ∅) ∧
  (∀a b. b ∈ interval [(a,b)] ⇔ interval [(a,b)] ≠ ∅) ∧
  (∀a b. a ∉ interval (a,b)) ∧ ∀a b. b ∉ interval (a,b)

EMPTY_INTERIOR_FINITE

⊢ ∀s. FINITE s ⇒ (interior s = ∅)

EMPTY_BIGUNION

⊢ ∀s. (BIGUNION s = ∅) ⇔ ∀t. t ∈ s ⇒ (t = ∅)

EMPTY_AS_INTERVAL

⊢ ∅ = interval [(1,0)]

DISTANCE_ATTAINS_SUP

⊢ ∀s a. compact s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,y) ≤ dist (a,x)

DISTANCE_ATTAINS_INF

⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ dist (a,x) ≤ dist (a,y)

DIST_TRIANGLE_LT

⊢ ∀x y z e. dist (x,z) + dist (y,z) < e ⇒ dist (x,y) < e

DIST_TRIANGLE_LE

⊢ ∀x y z e. dist (x,z) + dist (y,z) ≤ e ⇒ dist (x,y) ≤ e

DIST_TRIANGLE_HALF_R

⊢ ∀x1 x2 y. dist (y,x1) < e / 2 ∧ dist (y,x2) < e / 2 ⇒ dist (x1,x2) < e

DIST_TRIANGLE_HALF_L

⊢ ∀x1 x2 y. dist (x1,y) < e / 2 ∧ dist (x2,y) < e / 2 ⇒ dist (x1,x2) < e

DIST_TRIANGLE_EQ

⊢ ∀x y z.
      (dist (x,z) = dist (x,y) + dist (y,z)) ⇔
      (abs (x − y) * (y − z) = abs (y − z) * (x − y))

DIST_TRIANGLE_ALT

⊢ ∀x y z. dist (y,z) ≤ dist (x,y) + dist (x,z)

DIST_TRIANGLE_ADD_HALF

⊢ ∀x x' y y'.
      dist (x,x') < e / 2 ∧ dist (y,y') < e / 2 ⇒ dist (x + y,x' + y') < e

DIST_TRIANGLE_ADD

⊢ ∀x x' y y'. dist (x + y,x' + y') ≤ dist (x,x') + dist (y,y')

DIST_TRIANGLE

⊢ ∀x y z. dist (x,z) ≤ dist (x,y) + dist (y,z)

DIST_SYM

⊢ ∀x y. dist (x,y) = dist (y,x)

DIST_REFL

⊢ ∀x. dist (x,x) = 0

DIST_POS_LT

⊢ ∀x y. x ≠ y ⇒ 0 < dist (x,y)

DIST_POS_LE

⊢ ∀x y. 0 ≤ dist (x,y)

DIST_NZ

⊢ ∀x y. x ≠ y ⇔ 0 < dist (x,y)

DIST_MUL

⊢ ∀x y c. dist (c * x,c * y) = abs c * dist (x,y)

DIST_MIDPOINT

⊢ ∀a b.
      (dist (a,midpoint (a,b)) = dist (a,b) / 2) ∧
      (dist (b,midpoint (a,b)) = dist (a,b) / 2) ∧
      (dist (midpoint (a,b),a) = dist (a,b) / 2) ∧
      (dist (midpoint (a,b),b) = dist (a,b) / 2)

DIST_LE_0

⊢ ∀x y. dist (x,y) ≤ 0 ⇔ (x = y)

DIST_IN_OPEN_SEGMENT

⊢ ∀a b x.
      x ∈ segment (a,b) ⇒ dist (x,a) < dist (a,b) ∧ dist (x,b) < dist (a,b)

DIST_IN_OPEN_CLOSED_SEGMENT

⊢ (∀a b x.
       x ∈ segment [(a,b)] ⇒ dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b)) ∧
  ∀a b x.
      x ∈ segment (a,b) ⇒ dist (x,a) < dist (a,b) ∧ dist (x,b) < dist (a,b)

DIST_IN_CLOSED_SEGMENT

⊢ ∀a b x.
      x ∈ segment [(a,b)] ⇒ dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b)

DIST_EQ_0

⊢ ∀x y. (dist (x,y) = 0) ⇔ (x = y)

DIST_EQ

⊢ ∀w x y z. (dist (w,x) = dist (y,z)) ⇔ (dist (w,x) pow 2 = dist (y,z) pow 2)

DIST_CLOSEST_POINT_LIPSCHITZ

⊢ ∀s x y.
      closed s ∧ s ≠ ∅ ⇒
      abs (dist (x,closest_point s x) − dist (y,closest_point s y)) ≤
      dist (x,y)

DIST_0

⊢ ∀x. (dist (x,0) = abs x) ∧ (dist (0,x) = abs x)

DISJOINT_INTERVAL

⊢ ∀a b c d.
      ((interval [(a,b)] ∩ interval [(c,d)] = ∅) ⇔
       b < a ∨ d < c ∨ b < c ∨ d < a) ∧
      ((interval [(a,b)] ∩ interval (c,d) = ∅) ⇔ b < a ∨ d ≤ c ∨ b ≤ c ∨ d ≤ a) ∧
      ((interval (a,b) ∩ interval [(c,d)] = ∅) ⇔ b ≤ a ∨ d < c ∨ b ≤ c ∨ d ≤ a) ∧
      ((interval (a,b) ∩ interval (c,d) = ∅) ⇔ b ≤ a ∨ d ≤ c ∨ b ≤ c ∨ d ≤ a)

DISCRETE_IMP_CLOSED

⊢ ∀s e. 0 < e ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ abs (y − x) < e ⇒ (y = x)) ⇒ closed s

DISCRETE_BOUNDED_IMP_FINITE

⊢ ∀s e.
      0 < e ∧ (∀x y. x ∈ s ∧ y ∈ s ∧ abs (y − x) < e ⇒ (y = x)) ∧ bounded s ⇒
      FINITE s

DINI

⊢ ∀f g s.
      compact s ∧ (∀n. f n continuous_on s) ∧ g continuous_on s ∧
      (∀x. x ∈ s ⇒ ((λn. f n x) --> g x) sequentially) ∧
      (∀n x. x ∈ s ⇒ f n x ≤ f (n + 1) x) ⇒
      ∀e.
          0 < e ⇒
          eventually (λn. ∀x. x ∈ s ⇒ abs (f n x − g x) < e) sequentially

DIM_UNIV

⊢ dim 𝕌(:real) = 1

DIM_UNIQUE

⊢ ∀v b. b ⊆ v ∧ v ⊆ span b ∧ independent b ∧ b HAS_SIZE n ⇒ (dim v = n)

DIM_SUBSTANDARD

⊢ dim {x | x = 0} = 0

DIM_SUBSET_UNIV

⊢ ∀s. dim s ≤ 1

DIM_SUBSET

⊢ ∀s t. s ⊆ t ⇒ dim s ≤ dim t

DIM_LE_CARD

⊢ ∀s. FINITE s ⇒ dim s ≤ CARD s

DIFF_CLOSURE_SUBSET

⊢ ∀s t. closure s DIFF closure t ⊆ closure (s DIFF t)

DIFF_BIGINTER

⊢ ∀u s. u DIFF BIGINTER s = BIGUNION {u DIFF t | t ∈ s}

DIAMETER_SUMS

⊢ ∀s t.
      bounded s ∧ bounded t ⇒
      diameter {x + y | x ∈ s ∧ y ∈ t} ≤ diameter s + diameter t

DIAMETER_SUBSET_CBALL_NONEMPTY

⊢ ∀s. bounded s ∧ s ≠ ∅ ⇒ ∃z. z ∈ s ∧ s ⊆ cball (z,diameter s)

DIAMETER_SUBSET_CBALL

⊢ ∀s. bounded s ⇒ ∃z. s ⊆ cball (z,diameter s)

DIAMETER_SUBSET

⊢ ∀s t. s ⊆ t ∧ bounded t ⇒ diameter s ≤ diameter t

DIAMETER_SING

⊢ ∀a. diameter {a} = 0

DIAMETER_POS_LE

⊢ ∀s. bounded s ⇒ 0 ≤ diameter s

DIAMETER_LINEAR_IMAGE

⊢ ∀f s.
      linear f ∧ (∀x. abs (f x) = abs x) ⇒ (diameter (IMAGE f s) = diameter s)

DIAMETER_LE

⊢ ∀s d.
      (s ≠ ∅ ∨ 0 ≤ d) ∧ (∀x y. x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ d) ⇒
      diameter s ≤ d

DIAMETER_INTERVAL

⊢ (∀a b.
       diameter (interval [(a,b)]) =
       if interval [(a,b)] = ∅ then 0 else abs (b − a)) ∧
  ∀a b.
      diameter (interval (a,b)) =
      if interval (a,b) = ∅ then 0 else abs (b − a)

DIAMETER_EQ_0

⊢ ∀s. bounded s ⇒ ((diameter s = 0) ⇔ (s = ∅) ∨ ∃a. s = {a})

DIAMETER_EMPTY

⊢ diameter ∅ = 0

DIAMETER_CLOSURE

⊢ ∀s. bounded s ⇒ (diameter (closure s) = diameter s)

DIAMETER_CBALL

⊢ ∀a r. diameter (cball (a,r)) = if r < 0 then 0 else 2 * r

DIAMETER_BOUNDED_BOUND

⊢ ∀s x y. bounded s ∧ x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ diameter s

DIAMETER_BOUNDED

⊢ ∀s.
      bounded s ⇒
      (∀x y. x ∈ s ∧ y ∈ s ⇒ abs (x − y) ≤ diameter s) ∧
      ∀d. 0 ≤ d ∧ d < diameter s ⇒ ∃x y. x ∈ s ∧ y ∈ s ∧ abs (x − y) > d

DIAMETER_BALL

⊢ ∀a r. diameter (ball (a,r)) = if r < 0 then 0 else 2 * r

DEPENDENT_MONO

⊢ ∀s t. dependent s ∧ s ⊆ t ⇒ dependent t

DEPENDENT_EXPLICIT

⊢ ∀p.
      dependent p ⇔
      ∃s u.
          FINITE s ∧ s ⊆ p ∧ (∃v. v ∈ s ∧ u v ≠ 0) ∧ (sum s (λv. u v * v) = 0)

DEPENDENT_CHOICE_FIXED

⊢ ∀P R a.
      P 0 a ∧ (∀n x. P n x ⇒ ∃y. P (SUC n) y ∧ R n x y) ⇒
      ∃f. (f 0 = a) ∧ (∀n. P n (f n)) ∧ ∀n. R n (f n) (f (SUC n))

DEPENDENT_CHOICE

⊢ ∀P R.
      (∃a. P 0 a) ∧ (∀n x. P n x ⇒ ∃y. P (SUC n) y ∧ R n x y) ⇒
      ∃f. (∀n. P n (f n)) ∧ ∀n. R n (f n) (f (SUC n))

DENSE_OPEN_INTER

⊢ ∀s t u.
      open_in (subtopology euclidean u) s ∧ t ⊆ u ∨
      open_in (subtopology euclidean u) t ∧ s ⊆ u ⇒
      (u ⊆ closure (s ∩ t) ⇔ u ⊆ closure s ∧ u ⊆ closure t)

DENSE_LIMIT_POINTS

⊢ ∀x. ({x | x limit_point_of s} = 𝕌(:real)) ⇔ (closure s = 𝕌(:real))

DENSE_IMP_PERFECT

⊢ ∀s. (closure s = 𝕌(:real)) ⇒ ∀x. x ∈ s ⇒ x limit_point_of s

DECREASING_CLOSED_NEST_SING

⊢ ∀s.
      (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ∧
      (∀e. 0 < e ⇒ ∃n. ∀x y. x ∈ s n ∧ y ∈ s n ⇒ dist (x,y) < e) ⇒
      ∃a. BIGINTER {t | (∃n. t = s n)} = {a}

DECREASING_CLOSED_NEST

⊢ ∀s.
      (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ∧
      (∀e. 0 < e ⇒ ∃n. ∀x y. x ∈ s n ∧ y ∈ s n ⇒ dist (x,y) < e) ⇒
      ∃a. ∀n. a ∈ s n

COUNTABLE_OPEN_INTERVAL

⊢ ∀a b. COUNTABLE (interval (a,b)) ⇔ (interval (a,b) = ∅)

CONVERGENT_IMP_CAUCHY

⊢ ∀s l. (s --> l) sequentially ⇒ cauchy s

CONVERGENT_IMP_BOUNDED

⊢ ∀s l. (s --> l) sequentially ⇒ bounded (IMAGE s 𝕌(:num))

CONVERGENT_EQ_CAUCHY

⊢ ∀s. (∃l. (s --> l) sequentially) ⇔ cauchy s

CONVERGENT_BOUNDED_MONOTONE

⊢ ∀s b.
      (∀n. abs (s n) ≤ b) ∧
      ((∀m n. m ≤ n ⇒ s m ≤ s n) ∨ ∀m n. m ≤ n ⇒ s n ≤ s m) ⇒
      ∃l. ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s n − l) < e

CONVERGENT_BOUNDED_INCREASING

⊢ ∀s b.
      (∀m n. m ≤ n ⇒ s m ≤ s n) ∧ (∀n. abs (s n) ≤ b) ⇒
      ∃l. ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s n − l) < e

CONTRACTION_IMP_CONTINUOUS_ON

⊢ ∀f. (∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) ≤ dist (x,y)) ⇒ f continuous_on s

CONTINUOUS_WITHIN_SUBSET

⊢ ∀f s t x.
      f continuous (at x within s) ∧ t ⊆ s ⇒ f continuous (at x within t)

CONTINUOUS_WITHIN_SEQUENTIALLY

⊢ ∀f s a.
      f continuous (at a within s) ⇔
      ∀x.
          (∀n. x n ∈ s) ∧ (x --> a) sequentially ⇒
          (f ∘ x --> f a) sequentially

CONTINUOUS_WITHIN_OPEN

⊢ ∀f x u.
      f continuous (at x within u) ⇔
      ∀t.
          open t ∧ f x ∈ t ⇒
          ∃s. open s ∧ x ∈ s ∧ ∀x'. x' ∈ s ∧ x' ∈ u ⇒ f x' ∈ t

CONTINUOUS_WITHIN_ID

⊢ ∀a s. (λx. x) continuous (at a within s)

CONTINUOUS_WITHIN_COMPOSE

⊢ ∀f g x s.
      f continuous (at x within s) ∧ g continuous (at (f x) within IMAGE f s) ⇒
      g ∘ f continuous (at x within s)

CONTINUOUS_WITHIN_COMPARISON

⊢ ∀f g s a.
      g continuous (at a within s) ∧
      (∀x. x ∈ s ⇒ dist (f a,f x) ≤ dist (g a,g x)) ⇒
      f continuous (at a within s)

CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL

⊢ ∀a s. closed s ∧ a ∉ s ⇒ f continuous (at a within s)

CONTINUOUS_WITHIN_BALL

⊢ ∀f s x.
      f continuous (at x within s) ⇔
      ∀e. 0 < e ⇒ ∃d. 0 < d ∧ IMAGE f (ball (x,d) ∩ s) ⊆ ball (f x,e)

CONTINUOUS_WITHIN_AVOID

⊢ ∀f x s a.
      f continuous (at x within s) ∧ x ∈ s ∧ f x ≠ a ⇒
      ∃e. 0 < e ∧ ∀y. y ∈ s ∧ dist (x,y) < e ⇒ f y ≠ a

CONTINUOUS_WITHIN

⊢ ∀f x. f continuous (at x within s) ⇔ (f --> f x) (at x within s)

continuous_within

⊢ f continuous (at x within s) ⇔
  ∀e. 0 < e ⇒ ∃d. 0 < d ∧ ∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e

CONTINUOUS_VMUL

⊢ ∀net c v. c continuous net ⇒ (λx. c x * v) continuous net

CONTINUOUS_UNIFORM_LIMIT

⊢ ∀net f g s.
      ¬trivial_limit net ∧ eventually (λn. f n continuous_on s) net ∧
      (∀e. 0 < e ⇒ eventually (λn. ∀x. x ∈ s ⇒ abs (f n x − g x) < e) net) ⇒
      g continuous_on s

CONTINUOUS_TRIVIAL_LIMIT

⊢ ∀f net. trivial_limit net ⇒ f continuous net

CONTINUOUS_TRANSFORM_WITHIN_SET_IMP

⊢ ∀f a s t.
      eventually (λx. x ∈ t ⇒ x ∈ s) (at a) ∧ f continuous (at a within s) ⇒
      f continuous (at a within t)

CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN

⊢ ∀f g s t a.
      open_in (subtopology euclidean t) s ∧ a ∈ s ∧
      (∀x. x ∈ s ⇒ (f x = g x)) ∧ f continuous (at a within t) ⇒
      g continuous (at a within t)

CONTINUOUS_TRANSFORM_WITHIN_OPEN

⊢ ∀f g s a.
      open s ∧ a ∈ s ∧ (∀x. x ∈ s ⇒ (f x = g x)) ∧ f continuous at a ⇒
      g continuous at a

CONTINUOUS_TRANSFORM_WITHIN

⊢ ∀f g s x d.
      0 < d ∧ x ∈ s ∧ (∀x'. x' ∈ s ∧ dist (x',x) < d ⇒ (f x' = g x')) ∧
      f continuous (at x within s) ⇒
      g continuous (at x within s)

CONTINUOUS_TRANSFORM_AT

⊢ ∀f g x d.
      0 < d ∧ (∀x'. dist (x',x) < d ⇒ (f x' = g x')) ∧ f continuous at x ⇒
      g continuous at x

CONTINUOUS_SUM

⊢ ∀net f s.
      FINITE s ∧ (∀a. a ∈ s ⇒ f a continuous net) ⇒
      (λx. sum s (λa. f a x)) continuous net

CONTINUOUS_SUB

⊢ ∀f g net.
      f continuous net ∧ g continuous net ⇒ (λx. f x − g x) continuous net

CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP

⊢ ∀f g s t.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ g continuous_on t ∧ IMAGE g t ⊆ s ∧
      (∀y. y ∈ t ⇒ (f (g y) = y)) ⇒
      ∀u.
          u ⊆ t ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
           open_in (subtopology euclidean t) u)

CONTINUOUS_PRODUCT

⊢ ∀net f t.
      FINITE t ∧ (∀i. i ∈ t ⇒ (λx. f x i) continuous net) ⇒
      (λx. product t (f x)) continuous net

CONTINUOUS_POW

⊢ ∀net f n. (λx. f x) continuous net ⇒ (λx. f x pow n) continuous net

CONTINUOUS_OPEN_PREIMAGE_UNIV

⊢ ∀f s. (∀x. f continuous at x) ∧ open s ⇒ open {x | f x ∈ s}

CONTINUOUS_OPEN_PREIMAGE

⊢ ∀f s t. f continuous_on s ∧ open s ∧ open t ⇒ open {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_OPEN_IN_PREIMAGE_GEN

⊢ ∀f s t u.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧ open_in (subtopology euclidean t) u ⇒
      open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u}

CONTINUOUS_OPEN_IN_PREIMAGE_EQ

⊢ ∀f s.
      f continuous_on s ⇔
      ∀t. open t ⇒ open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_OPEN_IN_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ open t ⇒
      open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_ON_VMUL

⊢ ∀s c v. c continuous_on s ⇒ (λx. c x * v) continuous_on s

CONTINUOUS_ON_UNION_OPEN

⊢ ∀f s t.
      open s ∧ open t ∧ f continuous_on s ∧ f continuous_on t ⇒
      f continuous_on s ∪ t

CONTINUOUS_ON_UNION_LOCAL_OPEN

⊢ ∀f s.
      open_in (subtopology euclidean (s ∪ t)) s ∧
      open_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
      f continuous_on t ⇒
      f continuous_on s ∪ t

CONTINUOUS_ON_UNION_LOCAL

⊢ ∀f s.
      closed_in (subtopology euclidean (s ∪ t)) s ∧
      closed_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
      f continuous_on t ⇒
      f continuous_on s ∪ t

CONTINUOUS_ON_UNION

⊢ ∀f s t.
      closed s ∧ closed t ∧ f continuous_on s ∧ f continuous_on t ⇒
      f continuous_on s ∪ t

CONTINUOUS_ON_SUM

⊢ ∀t f s.
      FINITE s ∧ (∀a. a ∈ s ⇒ f a continuous_on t) ⇒
      (λx. sum s (λa. f a x)) continuous_on t

CONTINUOUS_ON_SUBSET

⊢ ∀f s t. f continuous_on s ∧ t ⊆ s ⇒ f continuous_on t

CONTINUOUS_ON_SUB

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on s ⇒ (λx. f x − g x) continuous_on s

CONTINUOUS_ON_SING

⊢ ∀f a. f continuous_on {a}

CONTINUOUS_ON_SETDIST

⊢ ∀s t. (λy. setdist ({y},s)) continuous_on t

CONTINUOUS_ON_SEQUENTIALLY

⊢ ∀f s.
      f continuous_on s ⇔
      ∀x a.
          a ∈ s ∧ (∀n. x n ∈ s) ∧ (x --> a) sequentially ⇒
          (f ∘ x --> f a) sequentially

CONTINUOUS_ON_RANGE

⊢ ∀f s.
      f continuous_on s ⇔
      ∀x.
          x ∈ s ⇒
          ∀e.
              0 < e ⇒
              ∃d.
                  0 < d ∧
                  ∀x'. x' ∈ s ∧ abs (x' − x) < d ⇒ abs (f x' − f x) < e

CONTINUOUS_ON_PRODUCT

⊢ ∀f s t.
      FINITE t ∧ (∀i. i ∈ t ⇒ (λx. f x i) continuous_on s) ⇒
      (λx. product t (f x)) continuous_on s

CONTINUOUS_ON_POW

⊢ ∀f s n. (λx. f x) continuous_on s ⇒ (λx. f x pow n) continuous_on s

CONTINUOUS_ON_OPEN_GEN

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      (f continuous_on s ⇔
       ∀u.
           open_in (subtopology euclidean t) u ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u})

CONTINUOUS_ON_OPEN_AVOID

⊢ ∀f x s a.
      f continuous_on s ∧ open s ∧ x ∈ s ∧ f x ≠ a ⇒
      ∃e. 0 < e ∧ ∀y. dist (x,y) < e ⇒ f y ≠ a

CONTINUOUS_ON_OPEN

⊢ ∀f s.
      f continuous_on s ⇔
      ∀t.
          open_in (subtopology euclidean (IMAGE f s)) t ⇒
          open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_ON_NO_LIMPT

⊢ ∀f s. ¬(∃x. x limit_point_of s) ⇒ f continuous_on s

CONTINUOUS_ON_NEG

⊢ ∀f s. f continuous_on s ⇒ (λx. -f x) continuous_on s

CONTINUOUS_ON_MUL

⊢ ∀s c f.
      c continuous_on s ∧ f continuous_on s ⇒ (λx. c x * f x) continuous_on s

CONTINUOUS_ON_MIN

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on s ⇒
      (λx. min (f x) (g x)) continuous_on s

CONTINUOUS_ON_MAX

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on s ⇒
      (λx. max (f x) (g x)) continuous_on s

CONTINUOUS_ON_LIFT_DOT

⊢ ∀s. (λy. a * y) continuous_on s

CONTINUOUS_ON_INVERSE_OPEN_MAP

⊢ ∀f g s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧ (∀x. x ∈ s ⇒ (g (f x) = x)) ∧
      (∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t) (IMAGE f u)) ⇒
      g continuous_on t

CONTINUOUS_ON_INVERSE_CLOSED_MAP

⊢ ∀f g s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧ (∀x. x ∈ s ⇒ (g (f x) = x)) ∧
      (∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u)) ⇒
      g continuous_on t

CONTINUOUS_ON_INVERSE

⊢ ∀f g s.
      f continuous_on s ∧ compact s ∧ (∀x. x ∈ s ⇒ (g (f x) = x)) ⇒
      g continuous_on IMAGE f s

CONTINUOUS_ON_INV

⊢ ∀f s.
      f continuous_on s ∧ (∀x. x ∈ s ⇒ f x ≠ 0) ⇒ realinv ∘ f continuous_on s

CONTINUOUS_ON_INTERIOR

⊢ ∀f s x. f continuous_on s ∧ x ∈ interior s ⇒ f continuous at x

CONTINUOUS_ON_IMP_OPEN_IN

⊢ ∀f s t.
      f continuous_on s ∧ open_in (subtopology euclidean (IMAGE f s)) t ⇒
      open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_ON_IMP_CLOSED_IN

⊢ ∀f s t.
      f continuous_on s ∧ closed_in (subtopology euclidean (IMAGE f s)) t ⇒
      closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_ON_ID

⊢ ∀s. (λx. x) continuous_on s

CONTINUOUS_ON_FINITE

⊢ ∀f s. FINITE s ⇒ f continuous_on s

CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN

⊢ ∀f s. f continuous_on s ⇔ ∀x. x ∈ s ⇒ f continuous (at x within s)

CONTINUOUS_ON_EQ_CONTINUOUS_AT

⊢ ∀f s. open s ⇒ (f continuous_on s ⇔ ∀x. x ∈ s ⇒ f continuous at x)

CONTINUOUS_ON_EQ

⊢ ∀f g s. (∀x. x ∈ s ⇒ (f x = g x)) ∧ f continuous_on s ⇒ g continuous_on s

CONTINUOUS_ON_EMPTY

⊢ ∀f. f continuous_on ∅

CONTINUOUS_ON_DOT2

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on s ⇒ (λx. f x * g x) continuous_on s

CONTINUOUS_ON_DIST_CLOSEST_POINT

⊢ ∀s t. closed s ∧ s ≠ ∅ ⇒ (λx. dist (x,closest_point s x)) continuous_on t

CONTINUOUS_ON_DIST

⊢ ∀a s. (λx. dist (a,x)) continuous_on s

CONTINUOUS_ON_CONST

⊢ ∀s c. (λx. c) continuous_on s

CONTINUOUS_ON_COMPOSE_QUOTIENT

⊢ ∀f g s t u.
      IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧
      (∀v.
           v ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ v} ⇔
            open_in (subtopology euclidean t) v)) ∧ g ∘ f continuous_on s ⇒
      g continuous_on t

CONTINUOUS_ON_COMPOSE

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on IMAGE f s ⇒ g ∘ f continuous_on s

CONTINUOUS_ON_COMPONENTS_GEN

⊢ ∀f s.
      (∀c.
           c ∈ components s ⇒
           open_in (subtopology euclidean s) c ∧ f continuous_on c) ⇒
      f continuous_on s

CONTINUOUS_ON_COMPONENTS_FINITE

⊢ ∀f s.
      FINITE (components s) ∧ (∀c. c ∈ components s ⇒ f continuous_on c) ⇒
      f continuous_on s

CONTINUOUS_ON_COMPONENT_COMPOSE

⊢ ∀f s. f continuous_on s ⇒ (λx. f x) continuous_on s

CONTINUOUS_ON_CMUL

⊢ ∀f c s. f continuous_on s ⇒ (λx. c * f x) continuous_on s

CONTINUOUS_ON_CLOSURE_SEQUENTIALLY

⊢ ∀f s.
      f continuous_on closure s ⇔
      ∀x a.
          a ∈ closure s ∧ (∀n. x n ∈ s) ∧ (x --> a) sequentially ⇒
          (f ∘ x --> f a) sequentially

CONTINUOUS_ON_CLOSURE_COMPONENT_LE

⊢ ∀f s x b.
      f continuous_on closure s ∧ (∀y. y ∈ s ⇒ f y ≤ b) ∧ x ∈ closure s ⇒
      f x ≤ b

CONTINUOUS_ON_CLOSURE_COMPONENT_GE

⊢ ∀f s x b.
      f continuous_on closure s ∧ (∀y. y ∈ s ⇒ b ≤ f y) ∧ x ∈ closure s ⇒
      b ≤ f x

CONTINUOUS_ON_CLOSURE_ABS_LE

⊢ ∀f s x b.
      f continuous_on closure s ∧ (∀y. y ∈ s ⇒ abs (f y) ≤ b) ∧ x ∈ closure s ⇒
      abs (f x) ≤ b

CONTINUOUS_ON_CLOSURE

⊢ ∀f s.
      f continuous_on closure s ⇔
      ∀x e.
          x ∈ closure s ∧ 0 < e ⇒
          ∃d. 0 < d ∧ ∀y. y ∈ s ∧ dist (y,x) < d ⇒ dist (f y,f x) < e

CONTINUOUS_ON_CLOSED_GEN

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      (f continuous_on s ⇔
       ∀u.
           closed_in (subtopology euclidean t) u ⇒
           closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u})

CONTINUOUS_ON_CLOSED

⊢ ∀f s.
      f continuous_on s ⇔
      ∀t.
          closed_in (subtopology euclidean (IMAGE f s)) t ⇒
          closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_ON_CASES_OPEN

⊢ ∀P f g s t.
      open s ∧ open t ∧ f continuous_on s ∧ g continuous_on t ∧
      (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ (f x = g x)) ⇒
      (λx. if P x then f x else g x) continuous_on s ∪ t

CONTINUOUS_ON_CASES_LOCAL_OPEN

⊢ ∀P f g s t.
      open_in (subtopology euclidean (s ∪ t)) s ∧
      open_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
      g continuous_on t ∧ (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ (f x = g x)) ⇒
      (λx. if P x then f x else g x) continuous_on s ∪ t

CONTINUOUS_ON_CASES_LOCAL

⊢ ∀P f g s t.
      closed_in (subtopology euclidean (s ∪ t)) s ∧
      closed_in (subtopology euclidean (s ∪ t)) t ∧ f continuous_on s ∧
      g continuous_on t ∧ (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ (f x = g x)) ⇒
      (λx. if P x then f x else g x) continuous_on s ∪ t

CONTINUOUS_ON_CASES_LE

⊢ ∀f g h s a.
      f continuous_on {t | t ∈ s ∧ h t ≤ a} ∧
      g continuous_on {t | t ∈ s ∧ a ≤ h t} ∧ h continuous_on s ∧
      (∀t. t ∈ s ∧ (h t = a) ⇒ (f t = g t)) ⇒
      (λt. if h t ≤ a then f t else g t) continuous_on s

CONTINUOUS_ON_CASES_1

⊢ ∀f g s a.
      f continuous_on {t | t ∈ s ∧ t ≤ a} ∧
      g continuous_on {t | t ∈ s ∧ a ≤ t} ∧ (a ∈ s ⇒ (f a = g a)) ⇒
      (λt. if t ≤ a then f t else g t) continuous_on s

CONTINUOUS_ON_CASES

⊢ ∀P f g s t.
      closed s ∧ closed t ∧ f continuous_on s ∧ g continuous_on t ∧
      (∀x. x ∈ s ∧ ¬P x ∨ x ∈ t ∧ P x ⇒ (f x = g x)) ⇒
      (λx. if P x then f x else g x) continuous_on s ∪ t

CONTINUOUS_ON_AVOID

⊢ ∀f x s a.
      f continuous_on s ∧ x ∈ s ∧ f x ≠ a ⇒
      ∃e. 0 < e ∧ ∀y. y ∈ s ∧ dist (x,y) < e ⇒ f y ≠ a

CONTINUOUS_ON_ADD

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on s ⇒ (λx. f x + g x) continuous_on s

CONTINUOUS_ON_ABS_COMPOSE

⊢ ∀f s. f continuous_on s ⇒ (λx. abs (f x)) continuous_on s

CONTINUOUS_ON_ABS

⊢ ∀f s. f continuous_on s ⇒ (λx. abs (f x)) continuous_on s

CONTINUOUS_ON

⊢ ∀f s. f continuous_on s ⇔ ∀x. x ∈ s ⇒ (f --> f x) (at x within s)

CONTINUOUS_NEG

⊢ ∀f net. f continuous net ⇒ (λx. -f x) continuous net

CONTINUOUS_MUL

⊢ ∀net f c.
      c continuous net ∧ f continuous net ⇒ (λx. c x * f x) continuous net

CONTINUOUS_MIN

⊢ ∀f g net.
      f continuous net ∧ g continuous net ⇒
      (λx. min (f x) (g x)) continuous net

CONTINUOUS_MAX

⊢ ∀f g net.
      f continuous net ∧ g continuous net ⇒
      (λx. max (f x) (g x)) continuous net

CONTINUOUS_MAP_CLOSURES

⊢ ∀f. f continuous_on 𝕌(:real) ⇔ ∀s. IMAGE f (closure s) ⊆ closure (IMAGE f s)

CONTINUOUS_LEVELSET_OPEN_IN_CASES

⊢ ∀f s a.
      connected s ∧ f continuous_on s ∧
      open_in (subtopology euclidean s) {x | x ∈ s ∧ (f x = a)} ⇒
      (∀x. x ∈ s ⇒ f x ≠ a) ∨ ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_LEVELSET_OPEN_IN

⊢ ∀f s a.
      connected s ∧ f continuous_on s ∧
      open_in (subtopology euclidean s) {x | x ∈ s ∧ (f x = a)} ∧
      (∃x. x ∈ s ∧ (f x = a)) ⇒
      ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_LEVELSET_OPEN

⊢ ∀f s a.
      connected s ∧ f continuous_on s ∧ open {x | x ∈ s ∧ (f x = a)} ∧
      (∃x. x ∈ s ∧ (f x = a)) ⇒
      ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP

⊢ ∀f g s.
      f continuous_on s ∧ g continuous_on IMAGE f s ∧
      (∀x. x ∈ s ⇒ (g (f x) = x)) ⇒
      ∀u.
          u ⊆ IMAGE f s ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
           open_in (subtopology euclidean (IMAGE f s)) u)

CONTINUOUS_LE_ON_CLOSURE

⊢ ∀f s a.
      f continuous_on closure s ∧ (∀x. x ∈ s ⇒ f x ≤ a) ⇒
      ∀x. x ∈ closure s ⇒ f x ≤ a

CONTINUOUS_INV

⊢ ∀net f. f continuous net ∧ f (netlimit net) ≠ 0 ⇒ realinv ∘ f continuous net

CONTINUOUS_IMP_QUOTIENT_MAP

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧ compact s ⇒
      ∀u.
          u ⊆ t ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
           open_in (subtopology euclidean t) u)

CONTINUOUS_IMP_CLOSED_MAP

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧ compact s ⇒
      ∀u.
          closed_in (subtopology euclidean s) u ⇒
          closed_in (subtopology euclidean t) (IMAGE f u)

CONTINUOUS_GE_ON_CLOSURE

⊢ ∀f s a.
      f continuous_on closure s ∧ (∀x. x ∈ s ⇒ a ≤ f x) ⇒
      ∀x. x ∈ closure s ⇒ a ≤ f x

CONTINUOUS_FINITE_RANGE_CONSTANT_EQ

⊢ ∀s.
      connected s ⇔
      ∀f. f continuous_on s ∧ FINITE (IMAGE f s) ⇒ ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_FINITE_RANGE_CONSTANT

⊢ ∀f s.
      connected s ∧ f continuous_on s ∧ FINITE (IMAGE f s) ⇒
      ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DOT2

⊢ ∀net f g.
      f continuous net ∧ g continuous net ⇒ (λx. f x * g x) continuous net

CONTINUOUS_DISCRETE_RANGE_CONSTANT_EQ

⊢ ∀s.
      connected s ⇔
      ∀f.
          f continuous_on s ∧
          (∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
          ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DISCRETE_RANGE_CONSTANT

⊢ ∀f s.
      connected s ∧ f continuous_on s ∧
      (∀x. x ∈ s ⇒ ∃e. 0 < e ∧ ∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
      ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ

⊢ ∀s.
      connected s ⇔
      ∀f t.
          f continuous_on s ∧ IMAGE f s ⊆ t ∧
          (∀y. y ∈ t ⇒ (connected_component t y = {y})) ⇒
          ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DISCONNECTED_RANGE_CONSTANT

⊢ ∀f s.
      connected s ∧ f continuous_on s ∧ IMAGE f s ⊆ t ∧
      (∀y. y ∈ t ⇒ (connected_component t y = {y})) ⇒
      ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DISCONNECTED_DISCRETE_FINITE_RANGE_CONSTANT_EQ

⊢ (∀s.
       connected s ⇔
       ∀f t.
           f continuous_on s ∧ IMAGE f s ⊆ t ∧
           (∀y. y ∈ t ⇒ (connected_component t y = {y})) ⇒
           ∃a. ∀x. x ∈ s ⇒ (f x = a)) ∧
  (∀s.
       connected s ⇔
       ∀f.
           f continuous_on s ∧
           (∀x.
                x ∈ s ⇒
                ∃e. 0 < e ∧ ∀y. y ∈ s ∧ f y ≠ f x ⇒ e ≤ abs (f y − f x)) ⇒
           ∃a. ∀x. x ∈ s ⇒ (f x = a)) ∧
  ∀s.
      connected s ⇔
      ∀f. f continuous_on s ∧ FINITE (IMAGE f s) ⇒ ∃a. ∀x. x ∈ s ⇒ (f x = a)

CONTINUOUS_DIAMETER

⊢ ∀s e.
      bounded s ∧ s ≠ ∅ ∧ 0 < e ⇒
      ∃d.
          0 < d ∧
          ∀t.
              bounded t ∧ t ≠ ∅ ∧ hausdist (s,t) < d ⇒
              abs (diameter s − diameter t) < e

CONTINUOUS_CONSTANT_ON_CLOSURE

⊢ ∀f s a.
      f continuous_on closure s ∧ (∀x. x ∈ s ⇒ (f x = a)) ⇒
      ∀x. x ∈ closure s ⇒ (f x = a)

CONTINUOUS_CONST

⊢ ∀net c. (λx. c) continuous net

CONTINUOUS_COMPONENT_COMPOSE

⊢ ∀net f i. f continuous net ⇒ (λx. f x) continuous net

CONTINUOUS_CMUL

⊢ ∀f c net. f continuous net ⇒ (λx. c * f x) continuous net

CONTINUOUS_CLOSED_PREIMAGE_UNIV

⊢ ∀f s. (∀x. f continuous at x) ∧ closed s ⇒ closed {x | f x ∈ s}

CONTINUOUS_CLOSED_PREIMAGE_CONSTANT

⊢ ∀f s. f continuous_on s ∧ closed s ⇒ closed {x | x ∈ s ∧ (f x = a)}

CONTINUOUS_CLOSED_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ closed s ∧ closed t ⇒ closed {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_CLOSED_IN_PREIMAGE_GEN

⊢ ∀f s t u.
      f continuous_on s ∧ IMAGE f s ⊆ t ∧
      closed_in (subtopology euclidean t) u ⇒
      closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u}

CONTINUOUS_CLOSED_IN_PREIMAGE_EQ

⊢ ∀f s.
      f continuous_on s ⇔
      ∀t. closed t ⇒ closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT

⊢ ∀f s a.
      f continuous_on s ⇒
      closed_in (subtopology euclidean s) {x | x ∈ s ∧ (f x = a)}

CONTINUOUS_CLOSED_IN_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ closed t ⇒
      closed_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t}

CONTINUOUS_CLOSED_IMP_CAUCHY_CONTINUOUS

⊢ ∀f s.
      f continuous_on s ∧ closed s ⇒
      ∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)

CONTINUOUS_ATTAINS_SUP

⊢ ∀f s.
      compact s ∧ s ≠ ∅ ∧ f continuous_on s ⇒
      ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ f y ≤ f x

CONTINUOUS_ATTAINS_INF

⊢ ∀f s.
      compact s ∧ s ≠ ∅ ∧ f continuous_on s ⇒
      ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ f x ≤ f y

CONTINUOUS_AT_WITHIN_INV

⊢ ∀f s a.
      f continuous (at a within s) ∧ f a ≠ 0 ⇒
      realinv ∘ f continuous (at a within s)

CONTINUOUS_AT_WITHIN

⊢ ∀f x s. f continuous at x ⇒ f continuous (at x within s)

CONTINUOUS_AT_TRANSLATION

⊢ ∀a z f. f continuous at (a + z) ⇔ (λx. f (a + x)) continuous at z

CONTINUOUS_AT_SETDIST

⊢ ∀s x. (λy. setdist ({y},s)) continuous at x

CONTINUOUS_AT_SEQUENTIALLY

⊢ ∀f a.
      f continuous at a ⇔
      ∀x. (x --> a) sequentially ⇒ (f ∘ x --> f a) sequentially

CONTINUOUS_AT_RANGE

⊢ ∀f x.
      f continuous at x ⇔
      ∀e. 0 < e ⇒ ∃d. 0 < d ∧ ∀x'. abs (x' − x) < d ⇒ abs (f x' − f x) < e

CONTINUOUS_AT_OPEN

⊢ ∀f x.
      f continuous at x ⇔
      ∀t. open t ∧ f x ∈ t ⇒ ∃s. open s ∧ x ∈ s ∧ ∀x'. x' ∈ s ⇒ f x' ∈ t

CONTINUOUS_AT_LIFT_DOT

⊢ ∀a x. (λy. a * y) continuous at x

CONTINUOUS_AT_INV

⊢ ∀f a. f continuous at a ∧ f a ≠ 0 ⇒ realinv ∘ f continuous at a

CONTINUOUS_AT_IMP_CONTINUOUS_ON

⊢ ∀f s. (∀x. x ∈ s ⇒ f continuous at x) ⇒ f continuous_on s

CONTINUOUS_AT_ID

⊢ ∀a. (λx. x) continuous at a

CONTINUOUS_AT_DIST_CLOSEST_POINT

⊢ ∀s x. closed s ∧ s ≠ ∅ ⇒ (λx. dist (x,closest_point s x)) continuous at x

CONTINUOUS_AT_DIST

⊢ ∀a x. (λx. dist (a,x)) continuous at x

CONTINUOUS_AT_COMPOSE_EQ

⊢ ∀f g h.
      g continuous at x ∧ h continuous at (g x) ∧ (∀y. g (h y) = y) ∧
      (h (g x) = x) ⇒
      (f continuous at (g x) ⇔ (λx. f (g x)) continuous at x)

CONTINUOUS_AT_COMPOSE

⊢ ∀f g x. f continuous at x ∧ g continuous at (f x) ⇒ g ∘ f continuous at x

CONTINUOUS_AT_BALL

⊢ ∀f x.
      f continuous at x ⇔
      ∀e. 0 < e ⇒ ∃d. 0 < d ∧ IMAGE f (ball (x,d)) ⊆ ball (f x,e)

CONTINUOUS_AT_AVOID

⊢ ∀f x a.
      f continuous at x ∧ f x ≠ a ⇒ ∃e. 0 < e ∧ ∀y. dist (x,y) < e ⇒ f y ≠ a

CONTINUOUS_AT_ABS

⊢ ∀x. abs continuous at x

CONTINUOUS_AT

⊢ ∀f x. f continuous at x ⇔ (f --> f x) (at x)

continuous_at

⊢ f continuous at x ⇔
  ∀e. 0 < e ⇒ ∃d. 0 < d ∧ ∀x'. dist (x',x) < d ⇒ dist (f x',f x) < e

CONTINUOUS_AGREE_ON_CLOSURE

⊢ ∀g h.
      g continuous_on closure s ∧ h continuous_on closure s ∧
      (∀x. x ∈ s ⇒ (g x = h x)) ⇒
      ∀x. x ∈ closure s ⇒ (g x = h x)

CONTINUOUS_ADD

⊢ ∀f g net.
      f continuous net ∧ g continuous net ⇒ (λx. f x + g x) continuous net

CONTINUOUS_ABS_COMPOSE

⊢ ∀net f. f continuous net ⇒ (λx. abs (f x)) continuous net

CONTINUOUS_ABS

⊢ ∀f net. f continuous net ⇒ (λx. abs (f x)) continuous net

CONNECTED_UNIV

⊢ connected 𝕌(:real)

CONNECTED_UNION_STRONG

⊢ ∀s t. connected s ∧ connected t ∧ closure s ∩ t ≠ ∅ ⇒ connected (s ∪ t)

CONNECTED_UNION

⊢ ∀s t. connected s ∧ connected t ∧ s ∩ t ≠ ∅ ⇒ connected (s ∪ t)

CONNECTED_TRANSLATION_EQ

⊢ ∀a s. connected (IMAGE (λx. a + x) s) ⇔ connected s

CONNECTED_TRANSLATION

⊢ ∀a s. connected s ⇒ connected (IMAGE (λx. a + x) s)

CONNECTED_SUBSET_CLOPEN

⊢ ∀u s c.
      closed_in (subtopology euclidean u) s ∧
      open_in (subtopology euclidean u) s ∧ connected c ∧ c ⊆ u ∧ c ∩ s ≠ ∅ ⇒
      c ⊆ s

CONNECTED_SING

⊢ ∀a. connected {a}

CONNECTED_SEGMENT

⊢ (∀a b. connected (segment [(a,b)])) ∧ ∀a b. connected (segment (a,b))

CONNECTED_SCALING

⊢ ∀s c. connected s ⇒ connected (IMAGE (λx. c * x) s)

CONNECTED_REAL_LEMMA

⊢ ∀f a b e1 e2.
      a ≤ b ∧ f a ∈ e1 ∧ f b ∈ e2 ∧
      (∀e x.
           a ≤ x ∧ x ≤ b ∧ 0 < e ⇒
           ∃d. 0 < d ∧ ∀y. abs (y − x) < d ⇒ dist (f y,f x) < e) ∧
      (∀y. y ∈ e1 ⇒ ∃e. 0 < e ∧ ∀y'. dist (y',y) < e ⇒ y' ∈ e1) ∧
      (∀y. y ∈ e2 ⇒ ∃e. 0 < e ∧ ∀y'. dist (y',y) < e ⇒ y' ∈ e2) ∧
      ¬(∃x. a ≤ x ∧ x ≤ b ∧ f x ∈ e1 ∧ f x ∈ e2) ⇒
      ∃x. a ≤ x ∧ x ≤ b ∧ f x ∉ e1 ∧ f x ∉ e2

CONNECTED_OPEN_SET

⊢ ∀s.
      open s ⇒
      (connected s ⇔
       ¬∃e1 e2.
           open e1 ∧ open e2 ∧ e1 ≠ ∅ ∧ e2 ≠ ∅ ∧ (e1 ∪ e2 = s) ∧ (e1 ∩ e2 = ∅))

CONNECTED_OPEN_MONOTONE_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀c.
           open_in (subtopology euclidean s) c ⇒
           open_in (subtopology euclidean t) (IMAGE f c)) ∧
      (∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ (f x = y)}) ⇒
      ∀c. connected c ∧ c ⊆ t ⇒ connected {x | x ∈ s ∧ f x ∈ c}

CONNECTED_OPEN_IN_EQ

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          open_in (subtopology euclidean s) e1 ∧
          open_in (subtopology euclidean s) e2 ∧ (e1 ∪ e2 = s) ∧
          (e1 ∩ e2 = ∅) ∧ e1 ≠ ∅ ∧ e2 ≠ ∅

CONNECTED_OPEN_IN

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          open_in (subtopology euclidean s) e1 ∧
          open_in (subtopology euclidean s) e2 ∧ s ⊆ e1 ∪ e2 ∧ (e1 ∩ e2 = ∅) ∧
          e1 ≠ ∅ ∧ e2 ≠ ∅

CONNECTED_NEST_GEN

⊢ ∀s.
      (∀n. closed (s n) ∧ connected (s n)) ∧ (∃n. compact (s n)) ∧
      (∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
      connected (BIGINTER {s n | n ∈ 𝕌(:num)})

CONNECTED_NEST

⊢ ∀s.
      (∀n. compact (s n) ∧ connected (s n)) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
      connected (BIGINTER {s n | n ∈ 𝕌(:num)})

CONNECTED_NEGATIONS

⊢ ∀s. connected s ⇒ connected (IMAGE (λx. -x) s)

CONNECTED_MONOTONE_QUOTIENT_PREIMAGE_GEN

⊢ ∀f s t c.
      (IMAGE f s = t) ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ∧
      (∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ (f x = y)}) ∧
      (open_in (subtopology euclidean t) c ∨
       closed_in (subtopology euclidean t) c) ∧ connected c ⇒
      connected {x | x ∈ s ∧ f x ∈ c}

CONNECTED_MONOTONE_QUOTIENT_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀u.
           u ⊆ t ⇒
           (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ u} ⇔
            open_in (subtopology euclidean t) u)) ∧
      (∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ (f x = y)}) ∧ connected t ⇒
      connected s

CONNECTED_LINEAR_IMAGE

⊢ ∀f s. connected s ∧ linear f ⇒ connected (IMAGE f s)

CONNECTED_IVT_HYPERPLANE

⊢ ∀s x y a b.
      connected s ∧ x ∈ s ∧ y ∈ s ∧ a * x ≤ b ∧ b ≤ a * y ⇒
      ∃z. z ∈ s ∧ (a * z = b)

CONNECTED_IVT_COMPONENT

⊢ ∀s x y a. connected s ∧ x ∈ s ∧ y ∈ s ∧ x ≤ a ∧ a ≤ y ⇒ ∃z. z ∈ s ∧ (z = a)

CONNECTED_INTERMEDIATE_CLOSURE

⊢ ∀s t. connected s ∧ s ⊆ t ∧ t ⊆ closure s ⇒ connected t

CONNECTED_INTER_FRONTIER

⊢ ∀s t. connected s ∧ s ∩ t ≠ ∅ ∧ s DIFF t ≠ ∅ ⇒ s ∩ frontier t ≠ ∅

CONNECTED_INDUCTION_SIMPLE

⊢ ∀P s.
      connected s ∧
      (∀a.
           a ∈ s ⇒
           ∃t.
               open_in (subtopology euclidean s) t ∧ a ∈ t ∧
               ∀x y. x ∈ t ∧ y ∈ t ∧ P x ⇒ P y) ⇒
      ∀a b. a ∈ s ∧ b ∈ s ∧ P a ⇒ P b

CONNECTED_INDUCTION

⊢ ∀P Q s.
      connected s ∧
      (∀t a. open_in (subtopology euclidean s) t ∧ a ∈ t ⇒ ∃z. z ∈ t ∧ P z) ∧
      (∀a.
           a ∈ s ⇒
           ∃t.
               open_in (subtopology euclidean s) t ∧ a ∈ t ∧
               ∀x y. x ∈ t ∧ y ∈ t ∧ P x ∧ P y ∧ Q x ⇒ Q y) ⇒
      ∀a b. a ∈ s ∧ b ∈ s ∧ P a ∧ P b ∧ Q a ⇒ Q b

CONNECTED_IMP_PERFECT_CLOSED

⊢ ∀s x. connected s ∧ closed s ∧ ¬(∃a. s = {a}) ⇒ (x limit_point_of s ⇔ x ∈ s)

CONNECTED_IMP_PERFECT

⊢ ∀s x. connected s ∧ ¬(∃a. s = {a}) ∧ x ∈ s ⇒ x limit_point_of s

CONNECTED_IFF_CONNECTED_COMPONENT

⊢ ∀s. connected s ⇔ ∀x y. x ∈ s ∧ y ∈ s ⇒ connected_component s x y

CONNECTED_IFF_CONNECTABLE_POINTS

⊢ ∀s.
      connected s ⇔
      ∀a b. a ∈ s ∧ b ∈ s ⇒ ∃t. connected t ∧ t ⊆ s ∧ a ∈ t ∧ b ∈ t

CONNECTED_FROM_OPEN_UNION_AND_INTER

⊢ ∀s t.
      open s ∧ open t ∧ connected (s ∪ t) ∧ connected (s ∩ t) ⇒
      connected s ∧ connected t

CONNECTED_FROM_CLOSED_UNION_AND_INTER

⊢ ∀s t.
      closed s ∧ closed t ∧ connected (s ∪ t) ∧ connected (s ∩ t) ⇒
      connected s ∧ connected t

CONNECTED_EQUIVALENCE_RELATION_GEN

⊢ ∀P R s.
      connected s ∧ (∀x y. R x y ⇒ R y x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧
      (∀t a. open_in (subtopology euclidean s) t ∧ a ∈ t ⇒ ∃z. z ∈ t ∧ P z) ∧
      (∀a.
           a ∈ s ⇒
           ∃t.
               open_in (subtopology euclidean s) t ∧ a ∈ t ∧
               ∀x y. x ∈ t ∧ y ∈ t ∧ P x ∧ P y ⇒ R x y) ⇒
      ∀a b. a ∈ s ∧ b ∈ s ∧ P a ∧ P b ⇒ R a b

CONNECTED_EQUIVALENCE_RELATION

⊢ ∀R s.
      connected s ∧ (∀x y. R x y ⇒ R y x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧
      (∀a.
           a ∈ s ⇒
           ∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧ ∀x. x ∈ t ⇒ R a x) ⇒
      ∀a b. a ∈ s ∧ b ∈ s ⇒ R a b

CONNECTED_EQ_CONNECTED_COMPONENTS_EQ

⊢ ∀s. connected s ⇔ ∀c c'. c ∈ components s ∧ c' ∈ components s ⇒ (c = c')

CONNECTED_EQ_CONNECTED_COMPONENT_EQ

⊢ ∀s.
      connected s ⇔
      ∀x y.
          x ∈ s ∧ y ∈ s ⇒ (connected_component s x = connected_component s y)

CONNECTED_EQ_COMPONENTS_SUBSET_SING_EXISTS

⊢ ∀s. connected s ⇔ ∃a. components s ⊆ {a}

CONNECTED_EQ_COMPONENTS_SUBSET_SING

⊢ ∀s. connected s ⇔ components s ⊆ {s}

CONNECTED_EMPTY

⊢ connected ∅

CONNECTED_DISJOINT_BIGUNION_OPEN_UNIQUE

⊢ ∀f f'.
      pairwise DISJOINT f ∧ pairwise DISJOINT f' ∧
      (∀s. s ∈ f ⇒ open s ∧ connected s ∧ s ≠ ∅) ∧
      (∀s. s ∈ f' ⇒ open s ∧ connected s ∧ s ≠ ∅) ∧ (BIGUNION f = BIGUNION f') ⇒
      (f = f')

CONNECTED_DIFF_OPEN_FROM_CLOSED

⊢ ∀s t u.
      s ⊆ t ∧ t ⊆ u ∧ open s ∧ closed t ∧ connected u ∧ connected (t DIFF s) ⇒
      connected (u DIFF s)

CONNECTED_CONTINUOUS_IMAGE

⊢ ∀f s. f continuous_on s ∧ connected s ⇒ connected (IMAGE f s)

CONNECTED_CONNECTED_COMPONENT_SET

⊢ ∀s. connected s ⇔ ∀x. x ∈ s ⇒ (connected_component s x = s)

CONNECTED_CONNECTED_COMPONENT

⊢ ∀s x. connected (connected_component s x)

CONNECTED_COMPONENT_UNIV

⊢ ∀x. connected_component 𝕌(:real) x = 𝕌(:real)

CONNECTED_COMPONENT_UNIQUE

⊢ ∀s c x.
      x ∈ c ∧ c ⊆ s ∧ connected c ∧
      (∀c'. x ∈ c' ∧ c' ⊆ s ∧ connected c' ⇒ c' ⊆ c) ⇒
      (connected_component s x = c)

CONNECTED_COMPONENT_TRANS

⊢ ∀s x y.
      connected_component s x y ∧ connected_component s y z ⇒
      connected_component s x z

CONNECTED_COMPONENT_SYM_EQ

⊢ ∀s x y. connected_component s x y ⇔ connected_component s y x

CONNECTED_COMPONENT_SYM

⊢ ∀s x y. connected_component s x y ⇒ connected_component s y x

CONNECTED_COMPONENT_SUBSET

⊢ ∀s x. connected_component s x ⊆ s

CONNECTED_COMPONENT_SET

⊢ ∀s x.
      connected_component s x = {y | ∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t}

CONNECTED_COMPONENT_REFL_EQ

⊢ ∀s x. connected_component s x x ⇔ x ∈ s

CONNECTED_COMPONENT_REFL

⊢ ∀s x. x ∈ s ⇒ connected_component s x x

CONNECTED_COMPONENT_OVERLAP

⊢ ∀s a b.
      connected_component s a ∩ connected_component s b ≠ ∅ ⇔
      a ∈ s ∧ b ∈ s ∧ (connected_component s a = connected_component s b)

CONNECTED_COMPONENT_OF_SUBSET

⊢ ∀s t x. s ⊆ t ∧ connected_component s x y ⇒ connected_component t x y

CONNECTED_COMPONENT_NONOVERLAP

⊢ ∀s a b.
      (connected_component s a ∩ connected_component s b = ∅) ⇔
      a ∉ s ∨ b ∉ s ∨ connected_component s a ≠ connected_component s b

CONNECTED_COMPONENT_MONO

⊢ ∀s t x. s ⊆ t ⇒ connected_component s x ⊆ connected_component t x

CONNECTED_COMPONENT_MAXIMAL

⊢ ∀s t x. x ∈ t ∧ connected t ∧ t ⊆ s ⇒ t ⊆ connected_component s x

CONNECTED_COMPONENT_INTERMEDIATE_SUBSET

⊢ ∀t u a.
      connected_component u a ⊆ t ∧ t ⊆ u ⇒
      (connected_component t a = connected_component u a)

CONNECTED_COMPONENT_IN

⊢ ∀s x y. connected_component s x y ⇒ x ∈ s ∧ y ∈ s

CONNECTED_COMPONENT_IDEMP

⊢ ∀s x.
      connected_component (connected_component s x) x =
      connected_component s x

CONNECTED_COMPONENT_EQUIVALENCE_RELATION

⊢ ∀R s.
      (∀x y. R x y ⇒ R y x) ∧ (∀x y z. R x y ∧ R y z ⇒ R x z) ∧
      (∀a.
           a ∈ s ⇒
           ∃t. open_in (subtopology euclidean s) t ∧ a ∈ t ∧ ∀x. x ∈ t ⇒ R a x) ⇒
      ∀a b. connected_component s a b ⇒ R a b

CONNECTED_COMPONENT_EQ_UNIV

⊢ ∀s x. (connected_component s x = 𝕌(:real)) ⇔ (s = 𝕌(:real))

CONNECTED_COMPONENT_EQ_SELF

⊢ ∀s x. connected s ∧ x ∈ s ⇒ (connected_component s x = s)

CONNECTED_COMPONENT_EQ_EQ

⊢ ∀s x y.
      (connected_component s x = connected_component s y) ⇔
      x ∉ s ∧ y ∉ s ∨ x ∈ s ∧ y ∈ s ∧ connected_component s x y

CONNECTED_COMPONENT_EQ_EMPTY

⊢ ∀s x. (connected_component s x = ∅) ⇔ x ∉ s

CONNECTED_COMPONENT_EQ

⊢ ∀s x y.
      y ∈ connected_component s x ⇒
      (connected_component s y = connected_component s x)

CONNECTED_COMPONENT_EMPTY

⊢ ∀x. connected_component ∅ x = ∅

CONNECTED_COMPONENT_DISJOINT

⊢ ∀s a b.
      DISJOINT (connected_component s a) (connected_component s b) ⇔
      a ∉ connected_component s b

CONNECTED_COMPONENT_BIGUNION

⊢ ∀s x. connected_component s x = BIGUNION {t | connected t ∧ x ∈ t ∧ t ⊆ s}

CONNECTED_CLOSURE

⊢ ∀s. connected s ⇒ connected (closure s)

CONNECTED_CLOSED_SET

⊢ ∀s.
      closed s ⇒
      (connected s ⇔
       ¬∃e1 e2.
           closed e1 ∧ closed e2 ∧ e1 ≠ ∅ ∧ e2 ≠ ∅ ∧ (e1 ∪ e2 = s) ∧
           (e1 ∩ e2 = ∅))

CONNECTED_CLOSED_MONOTONE_PREIMAGE

⊢ ∀f s t.
      f continuous_on s ∧ (IMAGE f s = t) ∧
      (∀c.
           closed_in (subtopology euclidean s) c ⇒
           closed_in (subtopology euclidean t) (IMAGE f c)) ∧
      (∀y. y ∈ t ⇒ connected {x | x ∈ s ∧ (f x = y)}) ⇒
      ∀c. connected c ∧ c ⊆ t ⇒ connected {x | x ∈ s ∧ f x ∈ c}

CONNECTED_CLOSED_IN_EQ

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          closed_in (subtopology euclidean s) e1 ∧
          closed_in (subtopology euclidean s) e2 ∧ (e1 ∪ e2 = s) ∧
          (e1 ∩ e2 = ∅) ∧ e1 ≠ ∅ ∧ e2 ≠ ∅

CONNECTED_CLOSED_IN

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          closed_in (subtopology euclidean s) e1 ∧
          closed_in (subtopology euclidean s) e2 ∧ s ⊆ e1 ∪ e2 ∧
          (e1 ∩ e2 = ∅) ∧ e1 ≠ ∅ ∧ e2 ≠ ∅

CONNECTED_CLOSED

⊢ ∀s.
      connected s ⇔
      ¬∃e1 e2.
          closed e1 ∧ closed e2 ∧ s ⊆ e1 ∪ e2 ∧ (e1 ∩ e2 ∩ s = ∅) ∧
          e1 ∩ s ≠ ∅ ∧ e2 ∩ s ≠ ∅

CONNECTED_CLOPEN

⊢ ∀s.
      connected s ⇔
      ∀t.
          open_in (subtopology euclidean s) t ∧
          closed_in (subtopology euclidean s) t ⇒
          (t = ∅) ∨ (t = s)

CONNECTED_CHAIN_GEN

⊢ ∀f.
      (∀s. s ∈ f ⇒ closed s ∧ connected s) ∧ (∃s. s ∈ f ∧ compact s) ∧
      (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
      connected (BIGINTER f)

CONNECTED_CHAIN

⊢ ∀f.
      (∀s. s ∈ f ⇒ compact s ∧ connected s) ∧
      (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
      connected (BIGINTER f)

CONNECTED_BIGUNION

⊢ ∀P. (∀s. s ∈ P ⇒ connected s) ∧ BIGINTER P ≠ ∅ ⇒ connected (BIGUNION P)

CONDENSATION_POINT_OF_SUBSET

⊢ ∀x s t. x condensation_point_of s ∧ s ⊆ t ⇒ x condensation_point_of t

CONDENSATION_POINT_INFINITE_CBALL

⊢ ∀s x. x condensation_point_of s ⇔ ∀e. 0 < e ⇒ ¬COUNTABLE (s ∩ cball (x,e))

CONDENSATION_POINT_INFINITE_BALL_CBALL

⊢ (∀s x. x condensation_point_of s ⇔ ∀e. 0 < e ⇒ ¬COUNTABLE (s ∩ ball (x,e))) ∧
  ∀s x. x condensation_point_of s ⇔ ∀e. 0 < e ⇒ ¬COUNTABLE (s ∩ cball (x,e))

CONDENSATION_POINT_INFINITE_BALL

⊢ ∀s x. x condensation_point_of s ⇔ ∀e. 0 < e ⇒ ¬COUNTABLE (s ∩ ball (x,e))

CONDENSATION_POINT_IMP_LIMPT

⊢ ∀x s. x condensation_point_of s ⇒ x limit_point_of s

COMPONENTS_UNIV

⊢ components 𝕌(:real) = {𝕌(:real)}

COMPONENTS_UNIQUE_EQ

⊢ ∀s k.
      (components s = k) ⇔
      (BIGUNION k = s) ∧
      ∀c.
          c ∈ k ⇒
          connected c ∧ c ≠ ∅ ∧ ∀c'. connected c' ∧ c ⊆ c' ∧ c' ⊆ s ⇒ (c' = c)

COMPONENTS_UNIQUE

⊢ ∀s k.
      (BIGUNION k = s) ∧
      (∀c.
           c ∈ k ⇒
           connected c ∧ c ≠ ∅ ∧
           ∀c'. connected c' ∧ c ⊆ c' ∧ c' ⊆ s ⇒ (c' = c)) ⇒
      (components s = k)

COMPONENTS_NONOVERLAP

⊢ ∀s c c'. c ∈ components s ∧ c' ∈ components s ⇒ ((c ∩ c' = ∅) ⇔ c ≠ c')

COMPONENTS_MAXIMAL

⊢ ∀s t c. c ∈ components s ∧ connected t ∧ t ⊆ s ∧ c ∩ t ≠ ∅ ⇒ t ⊆ c

COMPONENTS_INTERMEDIATE_SUBSET

⊢ ∀s t u. s ∈ components u ∧ s ⊆ t ∧ t ⊆ u ⇒ s ∈ components t

COMPONENTS_EQ_SING_N_EXISTS

⊢ (∀s. (components s = {s}) ⇔ connected s ∧ s ≠ ∅) ∧
  ∀s. (∃a. components s = {a}) ⇔ connected s ∧ s ≠ ∅

COMPONENTS_EQ_SING_EXISTS

⊢ ∀s. (∃a. components s = {a}) ⇔ connected s ∧ s ≠ ∅

COMPONENTS_EQ_SING

⊢ ∀s. (components s = {s}) ⇔ connected s ∧ s ≠ ∅

COMPONENTS_EQ_EMPTY

⊢ ∀s. (components s = ∅) ⇔ (s = ∅)

COMPONENTS_EQ

⊢ ∀s c c'. c ∈ components s ∧ c' ∈ components s ⇒ ((c = c') ⇔ c ∩ c' ≠ ∅)

COMPONENTS_EMPTY

⊢ components ∅ = ∅

COMPLETE_UNIV

⊢ complete 𝕌(:real)

COMPLETE_ISOMETRIC_IMAGE

⊢ ∀f s e.
      0 < e ∧ subspace s ∧ linear f ∧ (∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x) ∧
      complete s ⇒
      complete (IMAGE f s)

COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (complete (IMAGE f s) ⇔ complete s)

COMPLETE_INJECTIVE_LINEAR_IMAGE

⊢ ∀f.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      ∀s. complete s ⇒ complete (IMAGE f s)

COMPLETE_EQ_CLOSED

⊢ ∀s. complete s ⇔ closed s

COMPLEMENT_CONNECTED_COMPONENT_BIGUNION

⊢ ∀s x.
      s DIFF connected_component s x =
      BIGUNION
        ({connected_component s y | y | y ∈ s} DELETE connected_component s x)

COMPACT_UNION

⊢ ∀s t. compact s ∧ compact t ⇒ compact (s ∪ t)

COMPACT_UNIFORMLY_EQUICONTINUOUS

⊢ ∀fs s.
      (∀x e.
           x ∈ s ∧ 0 < e ⇒
           ∃d.
               0 < d ∧
               ∀f x'. f ∈ fs ∧ x' ∈ s ∧ dist (x',x) < d ⇒ dist (f x',f x) < e) ∧
      compact s ⇒
      ∀e.
          0 < e ⇒
          ∃d.
              0 < d ∧
              ∀f x x'.
                  f ∈ fs ∧ x ∈ s ∧ x' ∈ s ∧ dist (x',x) < d ⇒
                  dist (f x',f x) < e

COMPACT_UNIFORMLY_CONTINUOUS

⊢ ∀f s. f continuous_on s ∧ compact s ⇒ f uniformly_continuous_on s

COMPACT_TRANSLATION_EQ

⊢ ∀a s. compact (IMAGE (λx. a + x) s) ⇔ compact s

COMPACT_TRANSLATION

⊢ ∀s a. compact s ⇒ compact (IMAGE (λx. a + x) s)

COMPACT_SPHERE

⊢ ∀a r. compact (sphere (a,r))

COMPACT_SING

⊢ ∀a. compact {a}

COMPACT_SEQUENCE_WITH_LIMIT

⊢ ∀f l. (f --> l) sequentially ⇒ compact (l INSERT IMAGE f 𝕌(:num))

COMPACT_SCALING

⊢ ∀s c. compact s ⇒ compact (IMAGE (λx. c * x) s)

COMPACT_REAL_LEMMA

⊢ ∀s b.
      (∀n. abs (s n) ≤ b) ⇒
      ∃l r.
          (∀m n. m < n ⇒ r m < r n) ∧
          ∀e. 0 < e ⇒ ∃N. ∀n. N ≤ n ⇒ abs (s (r n) − l) < e

COMPACT_NEST

⊢ ∀s.
      (∀n. compact (s n) ∧ s n ≠ ∅) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ⇒
      BIGINTER {s n | n ∈ 𝕌(:num)} ≠ ∅

COMPACT_NEGATIONS

⊢ ∀s. compact s ⇒ compact (IMAGE (λx. -x) s)

COMPACT_LINEAR_IMAGE

⊢ ∀f s. compact s ∧ linear f ⇒ compact (IMAGE f s)

COMPACT_LEMMA

⊢ ∀s.
      bounded s ∧ (∀n. x n ∈ s) ⇒
      ∃l r.
          (∀m n. m < n ⇒ r m < r n) ∧
          ∀e. 0 < e ⇒ ∃N. ∀n i. N ≤ n ⇒ abs (x (r n) − l) < e

COMPACT_INTERVAL_EQ

⊢ (∀a b. compact (interval [(a,b)])) ∧
  ∀a b. compact (interval (a,b)) ⇔ (interval (a,b) = ∅)

COMPACT_INTERVAL

⊢ ∀a b. compact (interval [(a,b)])

COMPACT_INTER_CLOSED

⊢ ∀s t. compact s ∧ closed t ⇒ compact (s ∩ t)

COMPACT_INTER

⊢ ∀s t. compact s ∧ compact t ⇒ compact (s ∩ t)

COMPACT_INSERT

⊢ ∀a s. compact s ⇒ compact (a INSERT s)

COMPACT_IMP_TOTALLY_BOUNDED

⊢ ∀s.
      compact s ⇒
      ∀e.
          0 < e ⇒
          ∃k. FINITE k ∧ k ⊆ s ∧ s ⊆ BIGUNION (IMAGE (λx. ball (x,e)) k)

COMPACT_IMP_HEINE_BOREL

⊢ ∀s.
      compact s ⇒
      ∀f.
          (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
          ∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'

COMPACT_IMP_FIP

⊢ ∀s f.
      compact s ∧ (∀t. t ∈ f ⇒ closed t) ∧
      (∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
      s ∩ BIGINTER f ≠ ∅

COMPACT_IMP_COMPLETE

⊢ ∀s. compact s ⇒ complete s

COMPACT_IMP_CLOSED

⊢ ∀s. compact s ⇒ closed s

COMPACT_IMP_BOUNDED

⊢ ∀s. compact s ⇒ bounded s

COMPACT_FRONTIER_BOUNDED

⊢ ∀s. bounded s ⇒ compact (frontier s)

COMPACT_FRONTIER

⊢ ∀s. compact s ⇒ compact (frontier s)

COMPACT_FIP

⊢ ∀f.
      (∀t. t ∈ f ⇒ compact t) ∧ (∀f'. FINITE f' ∧ f' ⊆ f ⇒ BIGINTER f' ≠ ∅) ⇒
      BIGINTER f ≠ ∅

COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY

⊢ ∀s.
      compact s ⇔
      ∀f.
          (∀t. t ∈ f ⇒ open_in (subtopology euclidean s) t) ∧ s ⊆ BIGUNION f ⇒
          ∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'

COMPACT_EQ_HEINE_BOREL

⊢ ∀s.
      compact s ⇔
      ∀f.
          (∀t. t ∈ f ⇒ open t) ∧ s ⊆ BIGUNION f ⇒
          ∃f'. f' ⊆ f ∧ FINITE f' ∧ s ⊆ BIGUNION f'

COMPACT_EQ_BOUNDED_CLOSED

⊢ ∀s. compact s ⇔ bounded s ∧ closed s

COMPACT_EQ_BOLZANO_WEIERSTRASS

⊢ ∀s. compact s ⇔ ∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t

COMPACT_EMPTY

⊢ compact ∅

COMPACT_DIFF

⊢ ∀s t. compact s ∧ open t ⇒ compact (s DIFF t)

COMPACT_CONTINUOUS_IMAGE_EQ

⊢ ∀f s.
      (∀x y. x ∈ s ∧ y ∈ s ∧ (f x = f y) ⇒ (x = y)) ⇒
      (f continuous_on s ⇔ ∀t. compact t ∧ t ⊆ s ⇒ compact (IMAGE f t))

COMPACT_CONTINUOUS_IMAGE

⊢ ∀f s. f continuous_on s ∧ compact s ⇒ compact (IMAGE f s)

COMPACT_COMPONENTS

⊢ ∀s c. compact s ∧ c ∈ components s ⇒ compact c

COMPACT_CLOSURE

⊢ ∀s. compact (closure s) ⇔ bounded s

COMPACT_CLOSED_SUMS

⊢ ∀s t. compact s ∧ closed t ⇒ closed {x + y | x ∈ s ∧ y ∈ t}

COMPACT_CLOSED_DIFFERENCES

⊢ ∀s t. compact s ∧ closed t ⇒ closed {x − y | x ∈ s ∧ y ∈ t}

COMPACT_CHAIN

⊢ ∀f.
      (∀s. s ∈ f ⇒ compact s ∧ s ≠ ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ⇒
      BIGINTER f ≠ ∅

COMPACT_CBALL

⊢ ∀x e. compact (cball (x,e))

COMPACT_BIGUNION

⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ compact t) ⇒ compact (BIGUNION s)

COMPACT_BIGINTER

⊢ ∀f. (∀s. s ∈ f ⇒ compact s) ∧ f ≠ ∅ ⇒ compact (BIGINTER f)

COMPACT_ATTAINS_SUP

⊢ ∀s. compact s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ y ≤ x

COMPACT_ATTAINS_INF

⊢ ∀s. compact s ∧ s ≠ ∅ ⇒ ∃x. x ∈ s ∧ ∀y. y ∈ s ⇒ x ≤ y

COMPACT_AFFINITY

⊢ ∀s a c. compact s ⇒ compact (IMAGE (λx. a + c * x) s)

COLLINEAR_TRIPLES

⊢ ∀s a b.
      a ≠ b ⇒
      (collinear (a INSERT b INSERT s) ⇔ ∀x. x ∈ s ⇒ collinear {a; b; x})

COLLINEAR_SUBSET

⊢ ∀s t. collinear t ∧ s ⊆ t ⇒ collinear s

COLLINEAR_SMALL

⊢ ∀s. FINITE s ∧ CARD s ≤ 2 ⇒ collinear s

COLLINEAR_SING

⊢ ∀x. collinear {x}

COLLINEAR_MIDPOINT

⊢ ∀a b. collinear {a; midpoint (a,b); b}

COLLINEAR_LEMMA_ALT

⊢ ∀x y. collinear {0; x; y} ⇔ (x = 0) ∨ ∃c. y = c * x

COLLINEAR_LEMMA

⊢ ∀x y. collinear {0; x; y} ⇔ (x = 0) ∨ (y = 0) ∨ ∃c. y = c * x

COLLINEAR_EMPTY

⊢ collinear ∅

COLLINEAR_DIST_IN_OPEN_SEGMENT

⊢ ∀a b x.
      collinear {x; a; b} ∧ dist (x,a) < dist (a,b) ∧ dist (x,b) < dist (a,b) ⇒
      x ∈ segment (a,b)

COLLINEAR_DIST_IN_CLOSED_SEGMENT

⊢ ∀a b x.
      collinear {x; a; b} ∧ dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b) ⇒
      x ∈ segment [(a,b)]

COLLINEAR_DIST_BETWEEN

⊢ ∀a b x.
      collinear {x; a; b} ∧ dist (x,a) ≤ dist (a,b) ∧ dist (x,b) ≤ dist (a,b) ⇒
      between x (a,b)

COLLINEAR_BETWEEN_CASES

⊢ ∀a b c.
      collinear {a; b; c} ⇔
      between a (b,c) ∨ between b (c,a) ∨ between c (a,b)

COLLINEAR_4_3

⊢ ∀a b c d.
      a ≠ b ⇒
      (collinear {a; b; c; d} ⇔ collinear {a; b; c} ∧ collinear {a; b; d})

COLLINEAR_3_TRANS

⊢ ∀a b c d.
      collinear {a; b; c} ∧ collinear {b; c; d} ∧ b ≠ c ⇒ collinear {a; b; d}

COLLINEAR_3_EXPAND

⊢ ∀a b c. collinear {a; b; c} ⇔ (a = c) ∨ ∃u. b = u * a + (1 − u) * c

COLLINEAR_3

⊢ ∀x y z. collinear {x; y; z} ⇔ collinear {0; x − y; z − y}

COLLINEAR_2

⊢ ∀x y. collinear {x; y}

COLLINEAR_1

⊢ ∀s. collinear s

COBOUNDED_INTER_UNBOUNDED

⊢ ∀s t. bounded (𝕌(:real) DIFF s) ∧ ¬bounded t ⇒ s ∩ t ≠ ∅

COBOUNDED_IMP_UNBOUNDED

⊢ ∀s. bounded (𝕌(:real) DIFF s) ⇒ ¬bounded s

CLOSURE_UNIV

⊢ closure 𝕌(:real) = 𝕌(:real)

CLOSURE_UNIQUE

⊢ ∀s t.
      s ⊆ t ∧ closed t ∧ (∀t'. s ⊆ t' ∧ closed t' ⇒ t ⊆ t') ⇒ (closure s = t)

CLOSURE_UNION_FRONTIER

⊢ ∀s. closure s = s ∪ frontier s

CLOSURE_UNION

⊢ ∀s t. closure (s ∪ t) = closure s ∪ closure t

CLOSURE_SUMS

⊢ ∀s t.
      bounded s ∨ bounded t ⇒
      (closure {x + y | x ∈ s ∧ y ∈ t} =
       {x + y | x ∈ closure s ∧ y ∈ closure t})

CLOSURE_SUBSET_EQ

⊢ ∀s. closure s ⊆ s ⇔ closed s

CLOSURE_SUBSET

⊢ ∀s. s ⊆ closure s

CLOSURE_SING

⊢ ∀x. closure {x} = {x}

CLOSURE_SEQUENTIAL

⊢ ∀s l. l ∈ closure s ⇔ ∃x. (∀n. x n ∈ s) ∧ (x --> l) sequentially

CLOSURE_OPEN_INTERVAL

⊢ ∀a b. interval (a,b) ≠ ∅ ⇒ (closure (interval (a,b)) = interval [(a,b)])

CLOSURE_OPEN_INTER_SUPERSET

⊢ ∀s t. open s ∧ s ⊆ closure t ⇒ (closure (s ∩ t) = closure s)

CLOSURE_OPEN_INTER_CLOSURE

⊢ ∀s t. open s ⇒ (closure (s ∩ closure t) = closure (s ∩ t))

CLOSURE_OPEN_IN_INTER_CLOSURE

⊢ ∀s t u.
      open_in (subtopology euclidean u) s ∧ t ⊆ u ⇒
      (closure (s ∩ closure t) = closure (s ∩ t))

CLOSURE_NONEMPTY_OPEN_INTER

⊢ ∀s x. x ∈ closure s ⇔ ∀t. x ∈ t ∧ open t ⇒ s ∩ t ≠ ∅

CLOSURE_NEGATIONS

⊢ ∀s. closure (IMAGE (λx. -x) s) = IMAGE (λx. -x) (closure s)

CLOSURE_MINIMAL_EQ

⊢ ∀s t. closed t ⇒ (closure s ⊆ t ⇔ s ⊆ t)

CLOSURE_MINIMAL

⊢ ∀s t. s ⊆ t ∧ closed t ⇒ closure s ⊆ t

CLOSURE_LINEAR_IMAGE_SUBSET

⊢ ∀f s. linear f ⇒ IMAGE f (closure s) ⊆ closure (IMAGE f s)

CLOSURE_INTERVAL

⊢ (∀a b. closure (interval [(a,b)]) = interval [(a,b)]) ∧
  ∀a b.
      closure (interval (a,b)) =
      if interval (a,b) = ∅ then ∅ else interval [(a,b)]

CLOSURE_INTERIOR_UNION_CLOSED

⊢ ∀s t.
      closed s ∧ closed t ⇒
      (closure (interior (s ∪ t)) =
       closure (interior s) ∪ closure (interior t))

CLOSURE_INTERIOR_IDEMP

⊢ ∀s. closure (interior (closure (interior s))) = closure (interior s)

CLOSURE_INTERIOR

⊢ ∀s. closure s = 𝕌(:real) DIFF interior (𝕌(:real) DIFF s)

CLOSURE_INTER_SUBSET

⊢ ∀s t. closure (s ∩ t) ⊆ closure s ∩ closure t

CLOSURE_INJECTIVE_LINEAR_IMAGE

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (closure (IMAGE f s) = IMAGE f (closure s))

CLOSURE_IMAGE_CLOSURE

⊢ ∀f s.
      f continuous_on closure s ⇒
      (closure (IMAGE f (closure s)) = closure (IMAGE f s))

CLOSURE_IMAGE_BOUNDED

⊢ ∀f s.
      f continuous_on closure s ∧ bounded s ⇒
      (closure (IMAGE f s) = IMAGE f (closure s))

CLOSURE_HYPERPLANE

⊢ ∀a b. closure {x | a * x = b} = {x | a * x = b}

CLOSURE_HULL

⊢ ∀s. closure s = closed hull s

CLOSURE_HALFSPACE_LT

⊢ ∀a b. a ≠ 0 ⇒ (closure {x | a * x < b} = {x | a * x ≤ b})

CLOSURE_HALFSPACE_GT

⊢ ∀a b. a ≠ 0 ⇒ (closure {x | a * x > b} = {x | a * x ≥ b})

CLOSURE_HALFSPACE_COMPONENT_LT

⊢ ∀a. closure {x | x < a} = {x | x ≤ a}

CLOSURE_HALFSPACE_COMPONENT_GT

⊢ ∀a. closure {x | x > a} = {x | x ≥ a}

CLOSURE_EQ_EMPTY

⊢ ∀s. (closure s = ∅) ⇔ (s = ∅)

CLOSURE_EQ

⊢ ∀s. (closure s = s) ⇔ closed s

CLOSURE_EMPTY

⊢ closure ∅ = ∅

CLOSURE_COMPLEMENT

⊢ ∀s. closure (𝕌(:real) DIFF s) = 𝕌(:real) DIFF interior s

CLOSURE_CLOSURE

⊢ ∀s. closure (closure s) = closure s

CLOSURE_CLOSED

⊢ ∀s. closed s ⇒ (closure s = s)

CLOSURE_BOUNDED_LINEAR_IMAGE

⊢ ∀f s. linear f ∧ bounded s ⇒ (closure (IMAGE f s) = IMAGE f (closure s))

CLOSURE_BIGUNION

⊢ ∀f. FINITE f ⇒ (closure (BIGUNION f) = BIGUNION {closure s | s ∈ f})

CLOSURE_BIGINTER_SUBSET

⊢ ∀f. closure (BIGINTER f) ⊆ BIGINTER (IMAGE closure f)

CLOSURE_BALL

⊢ ∀x e. 0 < e ⇒ (closure (ball (x,e)) = cball (x,e))

CLOSURE_APPROACHABLE

⊢ ∀x s. x ∈ closure s ⇔ ∀e. 0 < e ⇒ ∃y. y ∈ s ∧ dist (y,x) < e

CLOSEST_POINT_SELF

⊢ ∀s x. x ∈ s ⇒ (closest_point s x = x)

CLOSEST_POINT_REFL

⊢ ∀s x. closed s ∧ s ≠ ∅ ⇒ ((closest_point s x = x) ⇔ x ∈ s)

CLOSEST_POINT_LE

⊢ ∀s a x. closed s ∧ x ∈ s ⇒ dist (a,closest_point s a) ≤ dist (a,x)

CLOSEST_POINT_IN_SET

⊢ ∀s a. closed s ∧ s ≠ ∅ ⇒ closest_point s a ∈ s

CLOSEST_POINT_IN_INTERIOR

⊢ ∀s x. closed s ∧ s ≠ ∅ ⇒ (closest_point s x ∈ interior s ⇔ x ∈ interior s)

CLOSEST_POINT_IN_FRONTIER

⊢ ∀s x. closed s ∧ s ≠ ∅ ∧ x ∉ interior s ⇒ closest_point s x ∈ frontier s

CLOSEST_POINT_EXISTS

⊢ ∀s a.
      closed s ∧ s ≠ ∅ ⇒
      closest_point s a ∈ s ∧
      ∀y. y ∈ s ⇒ dist (a,closest_point s a) ≤ dist (a,y)

CLOSED_UNIV

⊢ closed 𝕌(:real)

CLOSED_UNION_COMPACT_SUBSETS

⊢ ∀s.
      closed s ⇒
      ∃f.
          (∀n. compact (f n)) ∧ (∀n. f n ⊆ s) ∧ (∀n. f n ⊆ f (n + 1)) ∧
          (BIGUNION {f n | n ∈ 𝕌(:num)} = s) ∧
          ∀k. compact k ∧ k ⊆ s ⇒ ∃N. ∀n. n ≥ N ⇒ k ⊆ f n

CLOSED_UNION

⊢ ∀s t. closed s ∧ closed t ⇒ closed (s ∪ t)

CLOSED_SUBSTANDARD

⊢ closed {x | x = 0}

CLOSED_SUBSET_EQ

⊢ ∀u s. closed s ⇒ (closed_in (subtopology euclidean u) s ⇔ s ⊆ u)

CLOSED_SUBSET

⊢ ∀s t. s ⊆ t ∧ closed s ⇒ closed_in (subtopology euclidean t) s

CLOSED_STANDARD_HYPERPLANE

⊢ ∀a. closed {x | x = a}

CLOSED_SPHERE

⊢ ∀a r. closed (sphere (a,r))

CLOSED_SING

⊢ ∀a. closed {a}

CLOSED_SEQUENTIAL_LIMITS

⊢ ∀s. closed s ⇔ ∀x l. (∀n. x n ∈ s) ∧ (x --> l) sequentially ⇒ l ∈ s

CLOSED_SEGMENT_LINEAR_IMAGE

⊢ ∀f a b. linear f ⇒ (segment [(f a,f b)] = IMAGE f (segment [(a,b)]))

CLOSED_SCALING

⊢ ∀s c. closed s ⇒ closed (IMAGE (λx. c * x) s)

CLOSED_POSITIVE_ORTHANT

⊢ closed {x | 0 ≤ x}

CLOSED_OPEN_INTERVAL

⊢ ∀a b. a ≤ b ⇒ (interval [(a,b)] = interval (a,b) ∪ {a; b})

CLOSED_NEGATIONS

⊢ ∀s. closed s ⇒ closed (IMAGE (λx. -x) s)

CLOSED_MAP_RESTRICT

⊢ ∀f s t t'.
      (∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u)) ∧ t' ⊆ t ⇒
      ∀u.
          closed_in (subtopology euclidean {x | x ∈ s ∧ f x ∈ t'}) u ⇒
          closed_in (subtopology euclidean t') (IMAGE f u)

CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_POINT

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀k.
            closed_in (subtopology euclidean s) k ⇒
            closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
       ∀u y.
           open_in (subtopology euclidean s) u ∧ y ∈ t ∧
           {x | x ∈ s ∧ (f x = y)} ⊆ u ⇒
           ∃v.
               open_in (subtopology euclidean t) v ∧ y ∈ v ∧
               {x | x ∈ s ∧ f x ∈ v} ⊆ u)

CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_EQ

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀k.
            closed_in (subtopology euclidean s) k ⇒
            closed_in (subtopology euclidean t) (IMAGE f k)) ⇔
       ∀u w.
           open_in (subtopology euclidean s) u ∧ w ⊆ t ∧
           {x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
           ∃v.
               open_in (subtopology euclidean t) v ∧ w ⊆ v ∧
               {x | x ∈ s ∧ f x ∈ v} ⊆ u)

CLOSED_MAP_OPEN_SUPERSET_PREIMAGE

⊢ ∀f s t u w.
      (∀k.
           closed_in (subtopology euclidean s) k ⇒
           closed_in (subtopology euclidean t) (IMAGE f k)) ∧
      open_in (subtopology euclidean s) u ∧ w ⊆ t ∧ {x | x ∈ s ∧ f x ∈ w} ⊆ u ⇒
      ∃v.
          open_in (subtopology euclidean t) v ∧ w ⊆ v ∧
          {x | x ∈ s ∧ f x ∈ v} ⊆ u

CLOSED_MAP_IMP_QUOTIENT_MAP

⊢ ∀f s.
      f continuous_on s ∧
      (∀t.
           closed_in (subtopology euclidean s) t ⇒
           closed_in (subtopology euclidean (IMAGE f s)) (IMAGE f t)) ⇒
      ∀t.
          t ⊆ IMAGE f s ⇒
          (open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ t} ⇔
           open_in (subtopology euclidean (IMAGE f s)) t)

CLOSED_MAP_IMP_OPEN_MAP

⊢ ∀f s t.
      (IMAGE f s = t) ∧
      (∀u.
           closed_in (subtopology euclidean s) u ⇒
           closed_in (subtopology euclidean t) (IMAGE f u)) ∧
      (∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean s) {x | x ∈ s ∧ f x ∈ IMAGE f u}) ⇒
      ∀u.
          open_in (subtopology euclidean s) u ⇒
          open_in (subtopology euclidean t) (IMAGE f u)

CLOSED_MAP_IFF_UPPER_HEMICONTINUOUS_PREIMAGE

⊢ ∀f s t.
      IMAGE f s ⊆ t ⇒
      ((∀u.
            closed_in (subtopology euclidean s) u ⇒
            closed_in (subtopology euclidean t) (IMAGE f u)) ⇔
       ∀u.
           open_in (subtopology euclidean s) u ⇒
           open_in (subtopology euclidean t)
             {y | y ∈ t ∧ {x | x ∈ s ∧ (f x = y)} ⊆ u})

CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE

⊢ ∀f g s t u.
      f continuous_on s ∧ (IMAGE f s = t) ∧ IMAGE g t ⊆ u ∧
      (∀k.
           closed_in (subtopology euclidean s) k ⇒
           closed_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
      ∀k.
          closed_in (subtopology euclidean t) k ⇒
          closed_in (subtopology euclidean u) (IMAGE g k)

CLOSED_MAP_FROM_COMPOSITION_INJECTIVE

⊢ ∀f g s t u.
      IMAGE f s ⊆ t ∧ IMAGE g t ⊆ u ∧ g continuous_on t ∧
      (∀x y. x ∈ t ∧ y ∈ t ∧ (g x = g y) ⇒ (x = y)) ∧
      (∀k.
           closed_in (subtopology euclidean s) k ⇒
           closed_in (subtopology euclidean u) (IMAGE (g ∘ f) k)) ⇒
      ∀k.
          closed_in (subtopology euclidean s) k ⇒
          closed_in (subtopology euclidean t) (IMAGE f k)

CLOSED_MAP_CLOSURES

⊢ ∀f.
      (∀s. closed s ⇒ closed (IMAGE f s)) ⇔
      ∀s. closure (IMAGE f s) ⊆ IMAGE f (closure s)

CLOSED_LIMPTS

⊢ ∀s. closed {x | x limit_point_of s}

CLOSED_LIMPT

⊢ ∀s. closed s ⇔ ∀x. x limit_point_of s ⇒ x ∈ s

CLOSED_INTERVAL_RIGHT

⊢ ∀a. closed {x | a ≤ x}

CLOSED_INTERVAL_LEFT

⊢ ∀b. closed {x | x ≤ b}

CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL

⊢ ∀a b.
      interval [(a,b)] ≠ ∅ ⇒
      (interval [(a,b)] =
       IMAGE (λx. a + x) (IMAGE (λx. @f. f = (b − a) * x) (interval [(0,1)])))

CLOSED_INTERVAL_EQ

⊢ (∀a b. closed (interval [(a,b)])) ∧
  ∀a b. closed (interval (a,b)) ⇔ (interval (a,b) = ∅)

CLOSED_INTERVAL

⊢ ∀a b. closed (interval [(a,b)])

CLOSED_INTER_COMPACT

⊢ ∀s t. closed s ∧ compact t ⇒ compact (s ∩ t)

CLOSED_INTER

⊢ ∀s t. closed s ∧ closed t ⇒ closed (s ∩ t)

CLOSED_INSERT

⊢ ∀a s. closed s ⇒ closed (a INSERT s)

CLOSED_INJECTIVE_LINEAR_IMAGE_EQ

⊢ ∀f s.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      (closed (IMAGE f s) ⇔ closed s)

CLOSED_INJECTIVE_LINEAR_IMAGE

⊢ ∀f.
      linear f ∧ (∀x y. (f x = f y) ⇒ (x = y)) ⇒
      ∀s. closed s ⇒ closed (IMAGE f s)

CLOSED_INJECTIVE_IMAGE_SUBSPACE

⊢ ∀f s.
      subspace s ∧ linear f ∧ (∀x. x ∈ s ∧ (f x = 0) ⇒ (x = 0)) ∧ closed s ⇒
      closed (IMAGE f s)

CLOSED_IN_TRANS_EQ

⊢ ∀s t.
      (∀u.
           closed_in (subtopology euclidean t) u ⇒
           closed_in (subtopology euclidean s) t) ⇔
      closed_in (subtopology euclidean s) t

CLOSED_IN_TRANS

⊢ ∀s t u.
      closed_in (subtopology euclidean t) s ∧
      closed_in (subtopology euclidean u) t ⇒
      closed_in (subtopology euclidean u) s

CLOSED_IN_SUBTOPOLOGY_UNION

⊢ ∀top s t u.
      closed_in (subtopology top t) s ∧ closed_in (subtopology top u) s ⇒
      closed_in (subtopology top (t ∪ u)) s

CLOSED_IN_SUBTOPOLOGY_REFL

⊢ ∀top u. closed_in (subtopology top u) u ⇔ u ⊆ topspace top

CLOSED_IN_SUBTOPOLOGY_EMPTY

⊢ ∀top s. closed_in (subtopology top ∅) s ⇔ (s = ∅)

CLOSED_IN_SUBTOPOLOGY

⊢ ∀top u s.
      closed_in (subtopology top u) s ⇔ ∃t. closed_in top t ∧ (s = t ∩ u)

CLOSED_IN_SUBSET_TRANS

⊢ ∀s t u.
      closed_in (subtopology euclidean u) s ∧ s ⊆ t ∧ t ⊆ u ⇒
      closed_in (subtopology euclidean t) s

CLOSED_IN_SING

⊢ ∀u x. closed_in (subtopology euclidean u) {x} ⇔ x ∈ u

CLOSED_IN_REFL

⊢ ∀s. closed_in (subtopology euclidean s) s

CLOSED_IN_LIMPT

⊢ ∀s t.
      closed_in (subtopology euclidean t) s ⇔
      s ⊆ t ∧ ∀x. x limit_point_of s ∧ x ∈ t ⇒ x ∈ s

CLOSED_IN_INTER_CLOSURE

⊢ ∀s t. closed_in (subtopology euclidean s) t ⇔ (s ∩ closure t = t)

CLOSED_IN_INTER_CLOSED

⊢ ∀s t u.
      closed_in (subtopology euclidean u) s ∧ closed t ⇒
      closed_in (subtopology euclidean u) (s ∩ t)

CLOSED_IN_IMP_SUBSET

⊢ ∀top s t. closed_in (subtopology top s) t ⇒ t ⊆ s

CLOSED_IN_CONNECTED_COMPONENT

⊢ ∀s x. closed_in (subtopology euclidean s) (connected_component s x)

CLOSED_IN_COMPONENT

⊢ ∀s c. c ∈ components s ⇒ closed_in (subtopology euclidean s) c

CLOSED_IN_COMPACT_EQ

⊢ ∀s t.
      compact s ⇒ (closed_in (subtopology euclidean s) t ⇔ compact t ∧ t ⊆ s)

CLOSED_IN_COMPACT

⊢ ∀s t. compact s ∧ closed_in (subtopology euclidean s) t ⇒ compact t

CLOSED_IN_CLOSED_TRANS

⊢ ∀s t. closed_in (subtopology euclidean t) s ∧ closed t ⇒ closed s

CLOSED_IN_CLOSED_INTER

⊢ ∀u s. closed s ⇒ closed_in (subtopology euclidean u) (u ∩ s)

CLOSED_IN_CLOSED_EQ

⊢ ∀s t. closed s ⇒ (closed_in (subtopology euclidean s) t ⇔ closed t ∧ t ⊆ s)

CLOSED_IN_CLOSED

⊢ ∀s u. closed_in (subtopology euclidean u) s ⇔ ∃t. closed t ∧ (s = u ∩ t)

CLOSED_IN

⊢ ∀s. closed s ⇔ closed_in euclidean s

CLOSED_IMP_LOCALLY_COMPACT

⊢ ∀s. closed s ⇒ locally compact s

CLOSED_IMP_FIP_COMPACT

⊢ ∀s f.
      closed s ∧ (∀t. t ∈ f ⇒ compact t) ∧
      (∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
      s ∩ BIGINTER f ≠ ∅

CLOSED_IMP_FIP

⊢ ∀s f.
      closed s ∧ (∀t. t ∈ f ⇒ closed t) ∧ (∃t. t ∈ f ∧ bounded t) ∧
      (∀f'. FINITE f' ∧ f' ⊆ f ⇒ s ∩ BIGINTER f' ≠ ∅) ⇒
      s ∩ BIGINTER f ≠ ∅

CLOSED_HYPERPLANE

⊢ ∀a b. closed {x | a * x = b}

CLOSED_HALFSPACE_LE

⊢ ∀a b. closed {x | a * x ≤ b}

CLOSED_HALFSPACE_GE

⊢ ∀a b. closed {x | a * x ≥ b}

CLOSED_HALFSPACE_COMPONENT_LE

⊢ ∀a. closed {x | x ≤ a}

CLOSED_HALFSPACE_COMPONENT_GE

⊢ ∀a. closed {x | x ≥ a}

CLOSED_FORALL_IN

⊢ ∀P Q. (∀a. P a ⇒ closed {x | Q a x}) ⇒ closed {x | (∀a. P a ⇒ Q a x)}

CLOSED_FORALL

⊢ ∀Q. (∀a. closed {x | Q a x}) ⇒ closed {x | (∀a. Q a x)}

CLOSED_FIP

⊢ ∀f.
      (∀t. t ∈ f ⇒ closed t) ∧ (∃t. t ∈ f ∧ bounded t) ∧
      (∀f'. FINITE f' ∧ f' ⊆ f ⇒ BIGINTER f' ≠ ∅) ⇒
      BIGINTER f ≠ ∅

CLOSED_EMPTY

⊢ closed ∅

CLOSED_DIFF_OPEN_INTERVAL

⊢ ∀a b.
      interval [(a,b)] DIFF interval (a,b) =
      if interval [(a,b)] = ∅ then ∅ else {a; b}

CLOSED_DIFF

⊢ ∀s t. closed s ∧ open t ⇒ closed (s DIFF t)

CLOSED_CONTAINS_SEQUENTIAL_LIMIT

⊢ ∀s x l. closed s ∧ (∀n. x n ∈ s) ∧ (x --> l) sequentially ⇒ l ∈ s

CLOSED_CONNECTED_COMPONENT

⊢ ∀s x. closed s ⇒ closed (connected_component s x)

CLOSED_COMPONENTS

⊢ ∀s c. closed s ∧ c ∈ components s ⇒ closed c

CLOSED_COMPACT_SUMS

⊢ ∀s t. closed s ∧ compact t ⇒ closed {x + y | x ∈ s ∧ y ∈ t}

CLOSED_COMPACT_DIFFERENCES

⊢ ∀s t. closed s ∧ compact t ⇒ closed {x − y | x ∈ s ∧ y ∈ t}

CLOSED_CLOSURE

⊢ ∀s. closed (closure s)

CLOSED_CBALL

⊢ ∀x e. closed (cball (x,e))

CLOSED_BIGUNION

⊢ ∀s. FINITE s ∧ (∀t. t ∈ s ⇒ closed t) ⇒ closed (BIGUNION s)

CLOSED_BIGINTER_COMPACT

⊢ ∀s. closed s ⇔ ∀e. compact (cball (0,e) ∩ s)

CLOSED_BIGINTER

⊢ ∀f. (∀s. s ∈ f ⇒ closed s) ⇒ closed (BIGINTER f)

CLOSED_AS_GDELTA

⊢ ∀s. closed s ⇒ gdelta s

CLOSED_APPROACHABLE

⊢ ∀x s. closed s ⇒ ((∀e. 0 < e ⇒ ∃y. y ∈ s ∧ dist (y,x) < e) ⇔ x ∈ s)

CLOSED

⊢ ∀s.
      closed s ⇔
      ∀x. (∀e. 0 < e ⇒ ∃x'. x' ∈ s ∧ x' ≠ x ∧ abs (x' − x) < e) ⇒ x ∈ s

CLOPEN_IN_COMPONENTS

⊢ ∀u s.
      closed_in (subtopology euclidean u) s ∧
      open_in (subtopology euclidean u) s ∧ connected s ∧ s ≠ ∅ ⇒
      s ∈ components u

CLOPEN_BIGUNION_COMPONENTS

⊢ ∀u s.
      closed_in (subtopology euclidean u) s ∧
      open_in (subtopology euclidean u) s ⇒
      ∃k. k ⊆ components u ∧ (s = BIGUNION k)

CLOPEN

⊢ ∀s. closed s ∧ open s ⇔ (s = ∅) ∨ (s = 𝕌(:real))

CENTRE_IN_CBALL

⊢ ∀x e. x ∈ cball (x,e) ⇔ 0 ≤ e

CENTRE_IN_BALL

⊢ ∀x e. x ∈ ball (x,e) ⇔ 0 < e

CBALL_TRIVIAL

⊢ ∀x. cball (x,0) = {x}

CBALL_TRANSLATION

⊢ ∀a x r. cball (a + x,r) = IMAGE (λy. a + y) (cball (x,r))

CBALL_SING

⊢ ∀x e. (e = 0) ⇒ (cball (x,e) = {x})

CBALL_SCALING

⊢ ∀c. 0 < c ⇒ ∀x r. cball (c * x,c * r) = IMAGE (λx. c * x) (cball (x,r))

CBALL_MIN_INTER

⊢ ∀x d e. cball (x,min d e) = cball (x,d) ∩ cball (x,e)

CBALL_MAX_UNION

⊢ ∀a r s. cball (a,max r s) = cball (a,r) ∪ cball (a,s)

CBALL_LINEAR_IMAGE

⊢ ∀f x r.
      linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
      (cball (f x,r) = IMAGE f (cball (x,r)))

CBALL_INTERVAL_0

⊢ ∀e. cball (0,e) = interval [(-e,e)]

CBALL_INTERVAL

⊢ ∀x e. cball (x,e) = interval [(x − e,x + e)]

CBALL_EQ_SING

⊢ ∀x e. (cball (x,e) = {x}) ⇔ (e = 0)

CBALL_EQ_EMPTY

⊢ ∀x e. (cball (x,e) = ∅) ⇔ e < 0

CBALL_EMPTY

⊢ ∀x e. e < 0 ⇒ (cball (x,e) = ∅)

CBALL_DIFF_SPHERE

⊢ ∀a r. cball (a,r) DIFF sphere (a,r) = ball (a,r)

CBALL_DIFF_BALL

⊢ ∀a r. cball (a,r) DIFF ball (a,r) = sphere (a,r)

CAUCHY_ISOMETRIC

⊢ ∀f s e x.
      0 < e ∧ subspace s ∧ linear f ∧ (∀x. x ∈ s ⇒ abs (f x) ≥ e * abs x) ∧
      (∀n. x n ∈ s) ∧ cauchy (f ∘ x) ⇒
      cauchy x

CAUCHY_IMP_BOUNDED

⊢ ∀s. cauchy s ⇒ bounded {y | (∃n. y = s n)}

CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA

⊢ ∀f s.
      (∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒
      ∀a x.
          (∀n. x n ∈ s) ∧ (x --> a) sequentially ⇒
          ∃l.
              (f ∘ x --> l) sequentially ∧
              ∀y.
                  (∀n. y n ∈ s) ∧ (y --> a) sequentially ⇒
                  (f ∘ y --> l) sequentially

CAUCHY_CONTINUOUS_IMP_CONTINUOUS

⊢ ∀f s. (∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒ f continuous_on s

CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE

⊢ ∀f s.
      (∀x. cauchy x ∧ (∀n. x n ∈ s) ⇒ cauchy (f ∘ x)) ⇒
      ∃g. g continuous_on closure s ∧ ∀x. x ∈ s ⇒ (g x = f x)

CAUCHY

⊢ ∀s. cauchy s ⇔ ∀e. 0 < e ⇒ ∃N. ∀n. n ≥ N ⇒ dist (s n,s N) < e

CARD_STDBASIS

⊢ CARD {1} = 1

CARD_GE_DIM_INDEPENDENT

⊢ ∀v b. b ⊆ v ∧ independent b ∧ dim v ≤ CARD b ⇒ v ⊆ span b

CARD_FRONTIER_INTERVAL

⊢ ∀s. is_interval s ⇒ FINITE (frontier s) ∧ CARD (frontier s) ≤ 2

CARD_EQ_REAL_IMP_UNCOUNTABLE

⊢ ∀s. s ≈ 𝕌(:real) ⇒ ¬COUNTABLE s

CARD_EQ_REAL

⊢ 𝕌(:real) ≈ 𝕌(:num -> bool)

CARD_EQ_OPEN

⊢ ∀s. open s ∧ s ≠ ∅ ⇒ s ≈ 𝕌(:real)

CARD_EQ_INTERVAL

⊢ (∀a b. interval (a,b) ≠ ∅ ⇒ interval [(a,b)] ≈ 𝕌(:real)) ∧
  ∀a b. interval (a,b) ≠ ∅ ⇒ interval (a,b) ≈ 𝕌(:real)

CARD_EQ_EUCLIDEAN

⊢ 𝕌(:real) ≈ 𝕌(:real)

CARD_EQ_CBALL

⊢ ∀a r. 0 < r ⇒ cball (a,r) ≈ 𝕌(:real)

CARD_EQ_BALL

⊢ ∀a r. 0 < r ⇒ ball (a,r) ≈ 𝕌(:real)

BOUNDS_LINEAR_0

⊢ ∀A B. (∀n. A * n ≤ B) ⇔ (A = 0)

BOUNDS_LINEAR

⊢ ∀A B C. (∀n. A * n ≤ B * n + C) ⇔ A ≤ B

BOUNDED_UNION

⊢ ∀s t. bounded (s ∪ t) ⇔ bounded s ∧ bounded t

BOUNDED_UNIFORMLY_CONTINUOUS_IMAGE

⊢ ∀f s. f uniformly_continuous_on s ∧ bounded s ⇒ bounded (IMAGE f s)

BOUNDED_TRANSLATION_EQ

⊢ ∀a s. bounded (IMAGE (λx. a + x) s) ⇔ bounded s

BOUNDED_TRANSLATION

⊢ ∀a s. bounded s ⇒ bounded (IMAGE (λx. a + x) s)

BOUNDED_SUMS_IMAGES

⊢ ∀f t s.
      FINITE s ∧ (∀a. a ∈ s ⇒ bounded {f x a | x ∈ t}) ⇒
      bounded {sum s (f x) | x ∈ t}

BOUNDED_SUMS_IMAGE

⊢ ∀f g t.
      bounded {f x | x ∈ t} ∧ bounded {g x | x ∈ t} ⇒
      bounded {f x + g x | x ∈ t}

BOUNDED_SUMS

⊢ ∀s t. bounded s ∧ bounded t ⇒ bounded {x + y | x ∈ s ∧ y ∈ t}

BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC

⊢ ∀s. bounded s ⇒ ∃a. s ⊆ interval (-a,a)

BOUNDED_SUBSET_OPEN_INTERVAL

⊢ ∀s. bounded s ⇒ ∃a b. s ⊆ interval (a,b)

BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC

⊢ ∀s. bounded s ⇒ ∃a. s ⊆ interval [(-a,a)]

BOUNDED_SUBSET_CLOSED_INTERVAL

⊢ ∀s. bounded s ⇒ ∃a b. s ⊆ interval [(a,b)]

BOUNDED_SUBSET_CBALL

⊢ ∀s x. bounded s ⇒ ∃r. 0 < r ∧ s ⊆ cball (x,r)

BOUNDED_SUBSET_BALL

⊢ ∀s x. bounded s ⇒ ∃r. 0 < r ∧ s ⊆ ball (x,r)

BOUNDED_SUBSET

⊢ ∀s t. bounded t ∧ s ⊆ t ⇒ bounded s

BOUNDED_SPHERE

⊢ ∀a r. bounded (sphere (a,r))

BOUNDED_SING

⊢ ∀a. bounded {a}

BOUNDED_SCALING

⊢ ∀c s. bounded s ⇒ bounded (IMAGE (λx. c * x) s)

BOUNDED_POS_LT

⊢ ∀s. bounded s ⇔ ∃b. 0 < b ∧ ∀x. x ∈ s ⇒ abs x < b

BOUNDED_POS

⊢ ∀s. bounded s ⇔ ∃b. 0 < b ∧ ∀x. x ∈ s ⇒ abs x ≤ b

BOUNDED_PARTIAL_SUMS

⊢ ∀f k.
      bounded {sum (k .. n) f | n ∈ 𝕌(:num)} ⇒
      bounded {sum (m .. n) f | m ∈ 𝕌(:num) ∧ n ∈ 𝕌(:num)}

BOUNDED_NEGATIONS

⊢ ∀s. bounded s ⇒ bounded (IMAGE (λx. -x) s)

BOUNDED_LINEAR_IMAGE

⊢ ∀f s. bounded s ∧ linear f ⇒ bounded (IMAGE f s)

BOUNDED_INTERVAL

⊢ (∀a b. bounded (interval [(a,b)])) ∧ ∀a b. bounded (interval (a,b))

BOUNDED_INTERIOR

⊢ ∀s. bounded s ⇒ bounded (interior s)

BOUNDED_INTER

⊢ ∀s t. bounded s ∨ bounded t ⇒ bounded (s ∩ t)

BOUNDED_INSERT

⊢ ∀x s. bounded (x INSERT s) ⇔ bounded s

BOUNDED_INCREASING_CONVERGENT

⊢ ∀s.
      bounded {s n | n ∈ 𝕌(:num)} ∧ (∀n. s n ≤ s (SUC n)) ⇒
      ∃l. (s --> l) sequentially

BOUNDED_HAS_SUP

⊢ ∀s.
      bounded s ∧ s ≠ ∅ ⇒
      (∀x. x ∈ s ⇒ x ≤ sup s) ∧ ∀b. (∀x. x ∈ s ⇒ x ≤ b) ⇒ sup s ≤ b

BOUNDED_HAS_INF

⊢ ∀s.
      bounded s ∧ s ≠ ∅ ⇒
      (∀x. x ∈ s ⇒ inf s ≤ x) ∧ ∀b. (∀x. x ∈ s ⇒ b ≤ x) ⇒ b ≤ inf s

BOUNDED_FRONTIER

⊢ ∀s. bounded s ⇒ bounded (frontier s)

BOUNDED_EQ_BOLZANO_WEIERSTRASS

⊢ ∀s. bounded s ⇔ ∀t. t ⊆ s ∧ INFINITE t ⇒ ∃x. x limit_point_of t

BOUNDED_EMPTY

⊢ bounded ∅

BOUNDED_DIFFS

⊢ ∀s t. bounded s ∧ bounded t ⇒ bounded {x − y | x ∈ s ∧ y ∈ t}

BOUNDED_DIFF

⊢ ∀s t. bounded s ⇒ bounded (s DIFF t)

BOUNDED_DECREASING_CONVERGENT

⊢ ∀s.
      bounded {s n | n ∈ 𝕌(:num)} ∧ (∀n. s (SUC n) ≤ s n) ⇒
      ∃l. (s --> l) sequentially

BOUNDED_COMPONENTWISE

⊢ ∀s. bounded s ⇔ bounded (IMAGE (λx. x) s)

BOUNDED_CLOSURE_EQ

⊢ ∀s. bounded (closure s) ⇔ bounded s

BOUNDED_CLOSURE

⊢ ∀s. bounded s ⇒ bounded (closure s)

BOUNDED_CLOSED_NEST

⊢ ∀s.
      (∀n. closed (s n)) ∧ (∀n. s n ≠ ∅) ∧ (∀m n. m ≤ n ⇒ s n ⊆ s m) ∧
      bounded (s 0) ⇒
      ∃a. ∀n. a ∈ s n

BOUNDED_CLOSED_INTERVAL

⊢ ∀a b. bounded (interval [(a,b)])

BOUNDED_CLOSED_IMP_COMPACT

⊢ ∀s. bounded s ∧ closed s ⇒ compact s

BOUNDED_CLOSED_CHAIN

⊢ ∀f b.
      (∀s. s ∈ f ⇒ closed s ∧ s ≠ ∅) ∧ (∀s t. s ∈ f ∧ t ∈ f ⇒ s ⊆ t ∨ t ⊆ s) ∧
      b ∈ f ∧ bounded b ⇒
      BIGINTER f ≠ ∅

BOUNDED_CBALL

⊢ ∀x e. bounded (cball (x,e))

BOUNDED_BIGUNION

⊢ ∀f. FINITE f ∧ (∀s. s ∈ f ⇒ bounded s) ⇒ bounded (BIGUNION f)

BOUNDED_BIGINTER

⊢ ∀f. (∃s. s ∈ f ∧ bounded s) ⇒ bounded (BIGINTER f)

BOUNDED_BALL

⊢ ∀x e. bounded (ball (x,e))

BOLZANO_WEIERSTRASS_IMP_CLOSED

⊢ ∀s. (∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x ∈ s ∧ x limit_point_of t) ⇒ closed s

BOLZANO_WEIERSTRASS_IMP_BOUNDED

⊢ ∀s. (∀t. INFINITE t ∧ t ⊆ s ⇒ ∃x. x limit_point_of t) ⇒ bounded s

BOLZANO_WEIERSTRASS_CONTRAPOS

⊢ ∀s t. compact s ∧ t ⊆ s ∧ (∀x. x ∈ s ⇒ ¬(x limit_point_of t)) ⇒ FINITE t

BOLZANO_WEIERSTRASS

⊢ ∀s. bounded s ∧ INFINITE s ⇒ ∃x. x limit_point_of s

BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE

⊢ ∀f g h s.
      f uniformly_continuous_on s ∧ g uniformly_continuous_on s ∧ bilinear h ∧
      bounded (IMAGE f s) ∧ bounded (IMAGE g s) ⇒
      (λx. h (f x) (g x)) uniformly_continuous_on s

BILINEAR_SWAP

⊢ ∀op. bilinear (λx y. op y x) ⇔ bilinear op

BILINEAR_SUM_PARTIAL_SUC

⊢ ∀f g h m n.
      bilinear h ⇒
      (sum (m .. n) (λk. h (f k) (g (k + 1) − g k)) =
       if m ≤ n then
         h (f (n + 1)) (g (n + 1)) − h (f m) (g m) −
         sum (m .. n) (λk. h (f (k + 1) − f k) (g (k + 1)))
       else 0)

BILINEAR_SUM_PARTIAL_PRE

⊢ ∀f g h m n.
      bilinear h ⇒
      (sum (m .. n) (λk. h (f k) (g k − g (k − 1))) =
       if m ≤ n then
         h (f (n + 1)) (g n) − h (f m) (g (m − 1)) −
         sum (m .. n) (λk. h (f (k + 1) − f k) (g k))
       else 0)

BILINEAR_SUM

⊢ ∀h.
      bilinear h ∧ FINITE s ∧ FINITE t ⇒
      (h (sum s f) (sum t g) = sum (s × t) (λ(i,j). h (f i) (g j)))

BILINEAR_RZERO

⊢ ∀h x. bilinear h ⇒ (h x 0 = 0)

BILINEAR_RSUB

⊢ ∀h x y z. bilinear h ⇒ (h x (y − z) = h x y − h x z)

BILINEAR_RNEG

⊢ ∀h x y. bilinear h ⇒ (h x (-y) = -h x y)

BILINEAR_RMUL

⊢ ∀h c x y. bilinear h ⇒ (h x (c * y) = c * h x y)

BILINEAR_RADD

⊢ ∀h x y z. bilinear h ⇒ (h x (y + z) = h x y + h x z)

BILINEAR_LZERO

⊢ ∀h x. bilinear h ⇒ (h 0 x = 0)

BILINEAR_LSUB

⊢ ∀h x y z. bilinear h ⇒ (h (x − y) z = h x z − h y z)

BILINEAR_LNEG

⊢ ∀h x y. bilinear h ⇒ (h (-x) y = -h x y)

BILINEAR_LMUL

⊢ ∀h c x y. bilinear h ⇒ (h (c * x) y = c * h x y)

BILINEAR_LADD

⊢ ∀h x y z. bilinear h ⇒ (h (x + y) z = h x z + h y z)

BILINEAR_DOT

⊢ bilinear (λx y. x * y)

BILINEAR_CONTINUOUS_ON_COMPOSE

⊢ ∀f g h s.
      f continuous_on s ∧ g continuous_on s ∧ bilinear h ⇒
      (λx. h (f x) (g x)) continuous_on s

BILINEAR_CONTINUOUS_COMPOSE

⊢ ∀net f g h.
      f continuous net ∧ g continuous net ∧ bilinear h ⇒
      (λx. h (f x) (g x)) continuous net

BILINEAR_BOUNDED_POS

⊢ ∀h. bilinear h ⇒ ∃B. 0 < B ∧ ∀x y. abs (h x y) ≤ B * abs x * abs y

BILINEAR_BOUNDED

⊢ ∀h. bilinear h ⇒ ∃B. ∀x y. abs (h x y) ≤ B * abs x * abs y

BIGUNION_MONO_IMAGE

⊢ (∀x. x ∈ s ⇒ f x ⊆ g x) ⇒ BIGUNION (IMAGE f s) ⊆ BIGUNION (IMAGE g s)

BIGUNION_MONO

⊢ (∀x. x ∈ s ⇒ ∃y. y ∈ t ∧ x ⊆ y) ⇒ BIGUNION s ⊆ BIGUNION t

BIGUNION_IMAGE

⊢ ∀f s. BIGUNION (IMAGE f s) = {y | ∃x. x ∈ s ∧ y ∈ f x}

BIGUNION_GSPEC

⊢ (∀P f. BIGUNION {f x | P x} = {a | ∃x. P x ∧ a ∈ f x}) ∧
  (∀P f. BIGUNION {f x y | P x y} = {a | ∃x y. P x y ∧ a ∈ f x y}) ∧
  ∀P f. BIGUNION {f x y z | P x y z} = {a | ∃x y z. P x y z ∧ a ∈ f x y z}

BIGUNION_DIFF

⊢ ∀s t. BIGUNION s DIFF t = BIGUNION {x DIFF t | x ∈ s}

BIGUNION_CONNECTED_COMPONENT

⊢ ∀s. BIGUNION {connected_component s x | x | x ∈ s} = s

BIGUNION_COMPONENTS

⊢ ∀u. u = BIGUNION (components u)

BIGUNION_BIGINTER

⊢ ∀s. BIGUNION s = 𝕌(:α) DIFF BIGINTER {𝕌(:α) DIFF t | t ∈ s}

BIGINTER_IMAGE

⊢ ∀f s. BIGINTER (IMAGE f s) = {y | ∀x. x ∈ s ⇒ y ∈ f x}

BIGINTER_GSPEC

⊢ (∀P f. BIGINTER {f x | P x} = {a | ∀x. P x ⇒ a ∈ f x}) ∧
  (∀P f. BIGINTER {f x y | P x y} = {a | ∀x y. P x y ⇒ a ∈ f x y}) ∧
  ∀P f. BIGINTER {f x y z | P x y z} = {a | ∀x y z. P x y z ⇒ a ∈ f x y z}

BIGINTER_BIGUNION

⊢ ∀s. BIGINTER s = 𝕌(:α) DIFF BIGUNION {𝕌(:α) DIFF t | t ∈ s}

BETWEEN_TRANS_2

⊢ ∀a b c d. between a (b,c) ∧ between d (a,b) ⇒ between a (c,d)

BETWEEN_TRANS

⊢ ∀a b c d. between a (b,c) ∧ between d (a,c) ⇒ between d (b,c)

BETWEEN_SYM

⊢ ∀a b x. between x (a,b) ⇔ between x (b,a)

BETWEEN_REFL_EQ

⊢ ∀a x. between x (a,a) ⇔ (x = a)

BETWEEN_REFL

⊢ ∀a b. between a (a,b) ∧ between b (a,b) ∧ between a (a,a)

BETWEEN_MIDPOINT

⊢ ∀a b. between (midpoint (a,b)) (a,b) ∧ between (midpoint (a,b)) (b,a)

BETWEEN_IN_SEGMENT

⊢ ∀x a b. between x (a,b) ⇔ x ∈ segment [(a,b)]

BETWEEN_IMP_COLLINEAR

⊢ ∀a b x. between x (a,b) ⇒ collinear {a; x; b}

BETWEEN_ANTISYM

⊢ ∀a b c. between a (b,c) ∧ between b (a,c) ⇒ (a = b)

BETWEEN_ABS

⊢ ∀a b x. between x (a,b) ⇔ (abs (x − a) * (b − x) = abs (b − x) * (x − a))

BASIS_HAS_SIZE_DIM

⊢ ∀v b. independent b ∧ (span b = v) ⇒ b HAS_SIZE dim v

BASIS_EXISTS

⊢ ∀v. ∃b. b ⊆ v ∧ independent b ∧ v ⊆ span b ∧ b HAS_SIZE dim v

BASIS_CARD_EQ_DIM

⊢ ∀v b. b ⊆ v ∧ v ⊆ span b ∧ independent b ⇒ FINITE b ∧ (CARD b = dim v)

BANACH_FIX

⊢ ∀f s c.
      complete s ∧ s ≠ ∅ ∧ 0 ≤ c ∧ c < 1 ∧ IMAGE f s ⊆ s ∧
      (∀x y. x ∈ s ∧ y ∈ s ⇒ dist (f x,f y) ≤ c * dist (x,y)) ⇒
      ∃!x. x ∈ s ∧ (f x = x)

BALL_UNION_SPHERE

⊢ ∀a r. ball (a,r) ∪ sphere (a,r) = cball (a,r)

BALL_TRIVIAL

⊢ ∀x. ball (x,0) = ∅

BALL_TRANSLATION

⊢ ∀a x r. ball (a + x,r) = IMAGE (λy. a + y) (ball (x,r))

BALL_SUBSET_CBALL

⊢ ∀x e. ball (x,e) ⊆ cball (x,e)

BALL_SCALING

⊢ ∀c. 0 < c ⇒ ∀x r. ball (c * x,c * r) = IMAGE (λx. c * x) (ball (x,r))

BALL_MIN_INTER

⊢ ∀a r s. ball (a,min r s) = ball (a,r) ∩ ball (a,s)

BALL_MAX_UNION

⊢ ∀a r s. ball (a,max r s) = ball (a,r) ∪ ball (a,s)

BALL_LINEAR_IMAGE

⊢ ∀f x r.
      linear f ∧ (∀y. ∃x. f x = y) ∧ (∀x. abs (f x) = abs x) ⇒
      (ball (f x,r) = IMAGE f (ball (x,r)))

BALL_INTERVAL_0

⊢ ∀e. ball (0,e) = interval (-e,e)

BALL_INTERVAL

⊢ ∀x e. ball (x,e) = interval (x − e,x + e)

BALL_EQ_EMPTY

⊢ ∀x e. (ball (x,e) = ∅) ⇔ e ≤ 0

BALL_EMPTY

⊢ ∀x e. e ≤ 0 ⇒ (ball (x,e) = ∅)

BALL

⊢ ∀x r.
      (cball (x,r) = interval [(x − r,x + r)]) ∧
      (ball (x,r) = interval (x − r,x + r))

BAIRE_ALT

⊢ ∀g s.
      locally compact s ∧ s ≠ ∅ ∧ COUNTABLE g ∧ (BIGUNION g = s) ⇒
      ∃t u. t ∈ g ∧ open_in (subtopology euclidean s) u ∧ u ⊆ closure t

BAIRE

⊢ ∀g s.
      locally compact s ∧ COUNTABLE g ∧
      (∀t. t ∈ g ⇒ open_in (subtopology euclidean s) t ∧ s ⊆ closure t) ⇒
      s ⊆ closure (BIGINTER g)

AT_POSINFINITY

⊢ ∀x y. netord at_posinfinity x y ⇔ x ≥ y

AT_NEGINFINITY

⊢ ∀x y. netord at_neginfinity x y ⇔ x ≤ y

AT_INFINITY

⊢ ∀x y. netord at_infinity x y ⇔ abs x ≥ abs y

AT

⊢ ∀a x y. netord (at a) x y ⇔ 0 < dist (x,a) ∧ dist (x,a) ≤ dist (y,a)

APPROACHABLE_LT_LE

⊢ ∀P f. (∃d. 0 < d ∧ ∀x. f x < d ⇒ P x) ⇔ ∃d. 0 < d ∧ ∀x. f x ≤ d ⇒ P x

ALWAYS_EVENTUALLY

⊢ (∀x. p x) ⇒ eventually p net

AFFINITY_INVERSES

⊢ ∀m c.
      m ≠ 0 ⇒
      ((λx. m * x + c) ∘ (λx. m⁻¹ * x + -(m⁻¹ * c)) = (λx. x)) ∧
      ((λx. m⁻¹ * x + -(m⁻¹ * c)) ∘ (λx. m * x + c) = (λx. x))

ADD_SUBR2

⊢ ∀m n. m − (m + n) = 0

ADD_SUBR

⊢ ∀m n. n − (m + n) = 0

ADD_SUB2

⊢ ∀m n. m + n − m = n

ABS_TRIANGLE_LE

⊢ ∀x y. abs x + abs y ≤ e ⇒ abs (x + y) ≤ e

ABS_TRIANGLE_EQ

⊢ ∀x y. (abs (x + y) = abs x + abs y) ⇔ (abs x * y = abs y * x)

ABS_SUM_TRIVIAL_LEMMA

⊢ ∀e.
      0 < e ⇒
      (P ⇒ abs (sum (s ∩ (m .. n)) f) < e ⇔
       P ⇒ n < m ∨ abs (sum (s ∩ (m .. n)) f) < e)

ABS_LE_0

⊢ ∀x. abs x ≤ 0 ⇔ (x = 0)

ABS_CAUCHY_SCHWARZ_EQUAL

⊢ ∀x y. (abs (x * y) = abs x * abs y) ⇔ collinear {0; x; y}

ABS_CAUCHY_SCHWARZ_EQ

⊢ ∀x y. (x * y = abs x * abs y) ⇔ (abs x * y = abs y * x)

ABS_CAUCHY_SCHWARZ_ABS_EQ

⊢ ∀x y.
      (abs (x * y) = abs x * abs y) ⇔
      (abs x * y = abs y * x) ∨ (abs x * y = -abs y * x)