P. 357 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 35601-35700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	cdleme20e 35601	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. , , , represent s₂, f(s), t₂, f(t). We show <f(s),s₂,s> and <f(t),t₂,t> are centrally perspective. (Contributed by NM, 17-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdleme20f 35602	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line. , , , represent s₂, f(s), t₂, f(t). We show <f(s),s₂,s> and <f(t),t₂,t> are axially perspective. (Contributed by NM, 17-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdleme20g 35603	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. , , , represent s₂, f(s), t₂, f(t). (Contributed by NM, 18-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdleme20h 35604	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. , , , represent s₂, f(s), t₂, f(t). (Contributed by NM, 18-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdleme20i 35605	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. , , , represent s₂, f(s), t₂, f(t). We show (f(s) $\/$ s₂) $/\$ (f(t) $\/$ t₂) p $\/$ q. (Contributed by NM, 18-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdleme20j 35606	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. , , , represent s₂, f(s), t₂, f(t). We show s₂ t₂. (Contributed by NM, 18-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme20k 35607	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line. , , , represent s₂, f(s), t₂, f(t). (Contributed by NM, 20-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme20l1 35608	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. , , , represent s₂, f(s), t₂, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	cdleme20l2 35609	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. , , , represent s₂, f(s), t₂, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdleme20l 35610	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. , , , represent s₂, f(s), t₂, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	cdleme20m 35611	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. , , , , , represent s₂, f(s), f_s(r), t₂, f(t), f_t(r) respectively. We prove that if r s $\/$ t and u s $\/$ t, then f_s(r) = f_t(r). (Contributed by NM, 20-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme20 35612	Combine cdleme19f 35596 and cdleme20m 35611 to eliminate $.\/$ condition. (Contributed by NM, 28-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme21a 35613	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21b 35614	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme21c 35615	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme21at 35616	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21ct 35617	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme21d 35618	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. , , , , , represent s₂, f(s), f_s(r), z₂, f(z), f_z(r) respectively. We prove f_s(r) = f_z(r). (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21e 35619	Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line. , , , , , represent s₂, f(s), f_s(r), z₂, f(z), f_z(r) respectively. We prove that if u s $\/$ z, then f_t(r) = f_z(r). (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21f 35620	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21g 35621	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21h 35622*	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	cdleme21i 35623*	Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	cdleme21j 35624*	Combine cdleme20 35612 and cdleme21i 35623 to eliminate $.\/$ condition. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme21 35625	Part of proof of Lemma E in [Crawley] p. 113, 3rd line on p. 115. , , , , , represent s₂, f(s), f_s(r), t₂, f(t), f_t(r) respectively. Combine cdleme18d 35582 and cdleme21j 35624 to eliminate existence condition, proving f_s(r) = f_t(r) with fewer conditions. (Contributed by NM, 29-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme21k 35626	Eliminate condition in cdleme21 35625. (Contributed by NM, 26-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme22aa 35627	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t $\/$ v = p $\/$ q implies v = u. (Contributed by NM, 2-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme22a 35628	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 3rd line on p. 115. Show that t $\/$ v = p $\/$ q implies v = u. (Contributed by NM, 30-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme22b 35629	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t $\/$ v =/= p $\/$ q and s p $\/$ q implies t p $\/$ q. (Contributed by NM, 2-Dec-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$

Theorem	cdleme22cN 35630	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t $\/$ v =/= p $\/$ q and s p $\/$ q implies v p $\/$ q. (Contributed by NM, 3-Dec-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme22d 35631	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 9th line on p. 115. (Contributed by NM, 4-Dec-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$

Theorem	cdleme22e 35632	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. , , represent f(z), f_z(s), f_z(t) respectively. When t $\/$ v = p $\/$ q, f_z(s) f_z(t) $\/$ v. (Contributed by NM, 6-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme22eALTN 35633	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. , , represent f(z), f_z(s), f_z(t) respectively. When t $\/$ v = p $\/$ q, f_z(s) f_z(t) $\/$ v. (Contributed by NM, 6-Dec-2012.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme22f 35634	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. , represent f(t), f_t(s) respectively. If s t $\/$ v, then f_t(s) f(t) $\/$ v. (Contributed by NM, 6-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme22f2 35635	Part of proof of Lemma E in [Crawley] p. 113. cdleme22f 35634 with s and t swapped (this case is not mentioned by them). If s t $\/$ v, then f(s) f_s(t) $\/$ v. (Contributed by NM, 7-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme22g 35636	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. , represent f(s), f(t) respectively. If s t $\/$ v and s p $\/$ q, then f(s) f(t) $\/$ v. (Contributed by NM, 6-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme23a 35637	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdleme23b 35638	Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdleme23c 35639	Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$

Theorem	cdleme24 35640*	Quantified version of cdleme21k 35626. (Contributed by NM, 26-Dec-2012.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme25a 35641*	Lemma for cdleme25b 35642. (Contributed by NM, 1-Jan-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$

Theorem	cdleme25b 35642*	Transform cdleme24 35640. TODO get rid of $d's on , (Contributed by NM, 1-Jan-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme25c 35643*	Transform cdleme25b 35642. (Contributed by NM, 1-Jan-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme25dN 35644*	Transform cdleme25c 35643. (Contributed by NM, 19-Jan-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$

Theorem	cdleme25cl 35645*	Show closure of the unique element in cdleme25c 35643. (Contributed by NM, 2-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme25cv 35646*	Change bound variables in cdleme25c 35643. (Contributed by NM, 2-Feb-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme26e 35647*	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. , , represent f(z), f_z(s), f_z(t) respectively. When t $\/$ v = p $\/$ q, f_z(s) f_z(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme26ee 35648*	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. , , represent f(z), f_z(s), f_z(t) respectively. When t $\/$ v = p $\/$ q, f_z(s) f_z(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	cdleme26eALTN 35649*	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. , , represent f(z), f_z(s), f_z(t) respectively. When t $\/$ v = p $\/$ q, f_z(s) f_z(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	cdleme26fALTN 35650*	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. , represent f(t), f_t(s) respectively. If t t $\/$ v, then f_t(s) f(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme26f 35651*	Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. , represent f(t), f_t(s) respectively. If t t $\/$ v, then f_t(s) f(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme26f2ALTN 35652*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 35650 with s and t swapped (this case is not mentioned by them). If s t $\/$ v, then f(s) f_s(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme26f2 35653*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26fALTN 35650 with s and t swapped (this case is not mentioned by them). If s t $\/$ v, then f(s) f_s(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme27cl 35654*	Part of proof of Lemma E in [Crawley] p. 113. Closure of . (Contributed by NM, 6-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme27a 35655*	Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 35651 with s and t swapped (this case is not mentioned by them). If s t $\/$ v, then f(s) f_s(t) $\/$ v. TODO: FIX COMMENT. (Contributed by NM, 3-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme27b 35656*	Lemma for cdleme27N 35657. (Contributed by NM, 3-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$

Theorem	cdleme27N 35657*	Part of proof of Lemma E in [Crawley] p. 113. Eliminate the antecedent in cdleme27a 35655. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$

Theorem	cdleme28a 35658*	Lemma for cdleme25b 35642. TODO: FIX COMMENT. (Contributed by NM, 4-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme28b 35659*	Lemma for cdleme25b 35642. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme28c 35660*	Part of proof of Lemma E in [Crawley] p. 113. Eliminate the antecedent in cdleme28b 35659. TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme28 35661*	Quantified version of cdleme28c 35660. (Compare cdleme24 35640.) (Contributed by NM, 7-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme29ex 35662*	Lemma for cdleme29b 35663. (Compare cdleme25a 35641.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdleme29b 35663*	Transform cdleme28 35661. (Compare cdleme25b 35642.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme29c 35664*	Transform cdleme28b 35659. (Compare cdleme25c 35643.) TODO: FIX COMMENT. (Contributed by NM, 8-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme29cl 35665*	Show closure of the unique element in cdleme28c 35660. (Contributed by NM, 8-Feb-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdleme30a 35666	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Feb-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdleme31so 35667*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
$/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdleme31sn 35668*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
$.\/$ $.\/$

Theorem	cdleme31sn1 35669*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
$/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme31se 35670*	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdleme31se2 35671*	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdleme31sc 35672*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdleme31sde 35673*	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$

Theorem	cdleme31snd 35674*	Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Theorem	cdleme31sdnN 35675*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Theorem	cdleme31sn1c 35676*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $/\$ $.\/$

Theorem	cdleme31sn2 35677*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
$.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$

Theorem	cdleme31fv 35678*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
$/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$

Theorem	cdleme31fv1 35679*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
$/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$

Theorem	cdleme31fv1s 35680*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
$/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$

Theorem	cdleme31fv2 35681*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)
$/\$ $/\$ $/\$

Theorem	cdleme31id 35682*	Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.)
$/\$ $/\$

Theorem	cdlemefrs29pre00 35683	*START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW* FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 35316. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdlemefrs29bpre0 35684*	TODO fix comment. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemefrs29bpre1 35685*	TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemefrs29cpre1 35686*	TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemefrs29clN 35687*	TODO: NOT USED? Show closure of the unique element in cdlemefrs29cpre1 35686. (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemefrs32fva 35688*	Part of proof of Lemma E in [Crawley] p. 113. Value of at an atom not under . TODO: FIX COMMENT. TODO: consolidate uses of lhpmat 35316 here and elsewhere, and presence/absence of $.\/$ term. Also, why can proof be shortened with cdleme29cl 35665? What is difference from cdlemefs27cl 35701? (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemefrs32fva1 35689*	Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	cdlemefr29exN 35690*	Lemma for cdlemefs29bpre1N 35705. (Compare cdleme25a 35641.) TODO: FIX COMMENT. TODO: IS THIS NEEDED? (Contributed by NM, 28-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

Theorem	cdlemefr27cl 35691	Part of proof of Lemma E in [Crawley] p. 113. Closure of . (Contributed by NM, 23-Mar-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Theorem	cdlemefr32sn2aw 35692*	Show that is an atom not under when $.\/$ . (Contributed by NM, 28-Mar-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Theorem	cdlemefr32snb 35693*	Show closure of . (Contributed by NM, 28-Mar-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemefr29bpre0N 35694*	TODO fix comment. (Contributed by NM, 28-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	cdlemefr29clN 35695*	Show closure of the unique element in cdleme29c 35664. TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdleme43frv1snN 35696*	Value of when $.\/$ . (Contributed by NM, 30-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$

Theorem	cdlemefr32fvaN 35697*	Part of proof of Lemma E in [Crawley] p. 113. Value of at an atom not under . TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemefr32fva1 35698*	Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemefr31fv1 35699*	Value of when $.\/$ . TODO This may be useful for shortening others that now use riotasv 34245 3d . TODO: FIX COMMENT. (Contributed by NM, 30-Mar-2013.)
$.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	cdlemefs29pre00N 35700	FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 35316. (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$

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7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42551

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