Theorem List for Intuitionistic Logic Explorer - 4501-4600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | rexxpf 4501* |
Version of rexxp 4498 with bound-variable hypotheses. (Contributed
by NM,
19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | iunxpf 4502* |
Indexed union on a cross product is equals a double indexed union. The
hypothesis specifies an implicit substitution. (Contributed by NM,
19-Dec-2008.)
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Theorem | opabbi2dv 4503* |
Deduce equality of a relation and an ordered-pair class builder.
Compare abbi2dv 2197. (Contributed by NM, 24-Feb-2014.)
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Theorem | relop 4504* |
A necessary and sufficient condition for a Kuratowski ordered pair to be
a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this
detail.)
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Theorem | ideqg 4505 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon,
27-Aug-2011.)
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Theorem | ideq 4506 |
For sets, the identity relation is the same as equality. (Contributed
by NM, 13-Aug-1995.)
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Theorem | ididg 4507 |
A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof
shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | issetid 4508 |
Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario
Carneiro, 26-Apr-2015.)
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Theorem | coss1 4509 |
Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
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Theorem | coss2 4510 |
Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
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Theorem | coeq1 4511 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq2 4512 |
Equality theorem for composition of two classes. (Contributed by NM,
3-Jan-1997.)
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Theorem | coeq1i 4513 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2i 4514 |
Equality inference for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq1d 4515 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq2d 4516 |
Equality deduction for composition of two classes. (Contributed by NM,
16-Nov-2000.)
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Theorem | coeq12i 4517 |
Equality inference for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | coeq12d 4518 |
Equality deduction for composition of two classes. (Contributed by FL,
7-Jun-2012.)
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Theorem | nfco 4519 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 1-Sep-1999.)
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Theorem | brcog 4520* |
Ordered pair membership in a composition. (Contributed by NM,
24-Feb-2015.)
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Theorem | opelco2g 4521* |
Ordered pair membership in a composition. (Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | brcogw 4522 |
Ordered pair membership in a composition. (Contributed by Thierry
Arnoux, 14-Jan-2018.)
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Theorem | eqbrrdva 4523* |
Deduction from extensionality principle for relations, given an
equivalence only on the relation's domain and range. (Contributed by
Thierry Arnoux, 28-Nov-2017.)
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Theorem | brco 4524* |
Binary relation on a composition. (Contributed by NM, 21-Sep-2004.)
(Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | opelco 4525* |
Ordered pair membership in a composition. (Contributed by NM,
27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
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Theorem | cnvss 4526 |
Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
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Theorem | cnveq 4527 |
Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
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Theorem | cnveqi 4528 |
Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
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Theorem | cnveqd 4529 |
Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
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Theorem | elcnv 4530* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 24-Mar-1998.)
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Theorem | elcnv2 4531* |
Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed
by NM, 11-Aug-2004.)
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Theorem | nfcnv 4532 |
Bound-variable hypothesis builder for converse. (Contributed by NM,
31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | opelcnvg 4533 |
Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.)
(Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | brcnvg 4534 |
The converse of a binary relation swaps arguments. Theorem 11 of [Suppes]
p. 61. (Contributed by NM, 10-Oct-2005.)
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Theorem | opelcnv 4535 |
Ordered-pair membership in converse. (Contributed by NM,
13-Aug-1995.)
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Theorem | brcnv 4536 |
The converse of a binary relation swaps arguments. Theorem 11 of
[Suppes] p. 61. (Contributed by NM,
13-Aug-1995.)
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Theorem | csbcnvg 4537 |
Move class substitution in and out of the converse of a function.
(Contributed by Thierry Arnoux, 8-Feb-2017.)
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Theorem | cnvco 4538 |
Distributive law of converse over class composition. Theorem 26 of
[Suppes] p. 64. (Contributed by NM,
19-Mar-1998.) (Proof shortened by
Andrew Salmon, 27-Aug-2011.)
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Theorem | cnvuni 4539* |
The converse of a class union is the (indexed) union of the converses of
its members. (Contributed by NM, 11-Aug-2004.)
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Theorem | dfdm3 4540* |
Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfrn2 4541* |
Alternate definition of range. Definition 4 of [Suppes] p. 60.
(Contributed by NM, 27-Dec-1996.)
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Theorem | dfrn3 4542* |
Alternate definition of range. Definition 6.5(2) of [TakeutiZaring]
p. 24. (Contributed by NM, 28-Dec-1996.)
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Theorem | elrn2g 4543* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | elrng 4544* |
Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
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Theorem | dfdm4 4545 |
Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
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Theorem | dfdmf 4546* |
Definition of domain, using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by NM, 8-Mar-1995.)
(Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | csbdmg 4547 |
Distribute proper substitution through the domain of a class.
(Contributed by Jim Kingdon, 8-Dec-2018.)
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Theorem | eldmg 4548* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by Mario
Carneiro, 9-Jul-2014.)
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Theorem | eldm2g 4549* |
Domain membership. Theorem 4 of [Suppes] p. 59.
(Contributed by NM,
27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
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Theorem | eldm 4550* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 2-Apr-2004.)
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Theorem | eldm2 4551* |
Membership in a domain. Theorem 4 of [Suppes]
p. 59. (Contributed by
NM, 1-Aug-1994.)
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Theorem | dmss 4552 |
Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeq 4553 |
Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | dmeqi 4554 |
Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | dmeqd 4555 |
Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
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Theorem | opeldm 4556 |
Membership of first of an ordered pair in a domain. (Contributed by NM,
30-Jul-1995.)
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Theorem | breldm 4557 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 30-Jul-1995.)
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Theorem | opeldmg 4558 |
Membership of first of an ordered pair in a domain. (Contributed by Jim
Kingdon, 9-Jul-2019.)
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Theorem | breldmg 4559 |
Membership of first of a binary relation in a domain. (Contributed by
NM, 21-Mar-2007.)
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Theorem | dmun 4560 |
The domain of a union is the union of domains. Exercise 56(a) of
[Enderton] p. 65. (Contributed by NM,
12-Aug-1994.) (Proof shortened
by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmin 4561 |
The domain of an intersection belong to the intersection of domains.
Theorem 6 of [Suppes] p. 60.
(Contributed by NM, 15-Sep-2004.)
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Theorem | dmiun 4562 |
The domain of an indexed union. (Contributed by Mario Carneiro,
26-Apr-2016.)
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Theorem | dmuni 4563* |
The domain of a union. Part of Exercise 8 of [Enderton] p. 41.
(Contributed by NM, 3-Feb-2004.)
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Theorem | dmopab 4564* |
The domain of a class of ordered pairs. (Contributed by NM,
16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
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Theorem | dmopabss 4565* |
Upper bound for the domain of a restricted class of ordered pairs.
(Contributed by NM, 31-Jan-2004.)
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Theorem | dmopab3 4566* |
The domain of a restricted class of ordered pairs. (Contributed by NM,
31-Jan-2004.)
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Theorem | dm0 4567 |
The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1]
p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmi 4568 |
The domain of the identity relation is the universe. (Contributed by
NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
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Theorem | dmv 4569 |
The domain of the universe is the universe. (Contributed by NM,
8-Aug-2003.)
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Theorem | dm0rn0 4570 |
An empty domain implies an empty range. (Contributed by NM,
21-May-1998.)
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Theorem | reldm0 4571 |
A relation is empty iff its domain is empty. (Contributed by NM,
15-Sep-2004.)
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Theorem | dmmrnm 4572* |
A domain is inhabited if and only if the range is inhabited.
(Contributed by Jim Kingdon, 15-Dec-2018.)
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Theorem | dmxpm 4573* |
The domain of a cross product. Part of Theorem 3.13(x) of [Monk1]
p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew
Salmon, 27-Aug-2011.)
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Theorem | dmxpinm 4574* |
The domain of the intersection of two square cross products. Unlike
dmin 4561, equality holds. (Contributed by NM,
29-Jan-2008.)
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Theorem | xpid11m 4575* |
The cross product of a class with itself is one-to-one. (Contributed by
Jim Kingdon, 8-Dec-2018.)
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Theorem | dmcnvcnv 4576 |
The domain of the double converse of a class (which doesn't have to be a
relation as in dfrel2 4791). (Contributed by NM, 8-Apr-2007.)
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Theorem | rncnvcnv 4577 |
The range of the double converse of a class. (Contributed by NM,
8-Apr-2007.)
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Theorem | elreldm 4578 |
The first member of an ordered pair in a relation belongs to the domain
of the relation. (Contributed by NM, 28-Jul-2004.)
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Theorem | rneq 4579 |
Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
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Theorem | rneqi 4580 |
Equality inference for range. (Contributed by NM, 4-Mar-2004.)
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Theorem | rneqd 4581 |
Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
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Theorem | rnss 4582 |
Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
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Theorem | brelrng 4583 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 29-Jun-2008.)
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Theorem | opelrng 4584 |
Membership of second member of an ordered pair in a range. (Contributed
by Jim Kingdon, 26-Jan-2019.)
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Theorem | brelrn 4585 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 13-Aug-2004.)
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Theorem | opelrn 4586 |
Membership of second member of an ordered pair in a range. (Contributed
by NM, 23-Feb-1997.)
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Theorem | releldm 4587 |
The first argument of a binary relation belongs to its domain.
(Contributed by NM, 2-Jul-2008.)
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Theorem | relelrn 4588 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 2-Jul-2008.)
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Theorem | releldmb 4589* |
Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
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Theorem | relelrnb 4590* |
Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
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Theorem | releldmi 4591 |
The first argument of a binary relation belongs to its domain.
(Contributed by NM, 28-Apr-2015.)
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Theorem | relelrni 4592 |
The second argument of a binary relation belongs to its range.
(Contributed by NM, 28-Apr-2015.)
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Theorem | dfrnf 4593* |
Definition of range, using bound-variable hypotheses instead of distinct
variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by
Mario Carneiro, 15-Oct-2016.)
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Theorem | elrn2 4594* |
Membership in a range. (Contributed by NM, 10-Jul-1994.)
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Theorem | elrn 4595* |
Membership in a range. (Contributed by NM, 2-Apr-2004.)
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Theorem | nfdm 4596 |
Bound-variable hypothesis builder for domain. (Contributed by NM,
30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | nfrn 4597 |
Bound-variable hypothesis builder for range. (Contributed by NM,
1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | dmiin 4598 |
Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
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Theorem | rnopab 4599* |
The range of a class of ordered pairs. (Contributed by NM,
14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
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Theorem | rnmpt 4600* |
The range of a function in maps-to notation. (Contributed by Scott
Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
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