Library GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric

Require Export GeoCoq.Elements.OriginalProofs.euclidean_tactics.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_parallelsymmetric :
    A B C D,
   Par A B C D
   Par C D A B.
Proof.
intros.
let Tf:=fresh in
assert (Tf: a b c d m, (neq A B neq C D Col A B a Col A B b neq a b Col C D c Col C D d neq c d ¬ Meet A B C D BetS a m d BetS c m b)) by (conclude_def Par );destruct Tf as [a[b[c[d[m]]]]];spliter.
assert (¬ Meet C D A B).
 {
 intro.
 let Tf:=fresh in
 assert (Tf: P, (neq C D neq A B Col C D P Col A B P)) by (conclude_def Meet );destruct Tf as [P];spliter.
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (Par C D A B) by (conclude_def Par ).
close.
Qed.

End Euclid.