Library GeoCoq.Elements.OriginalProofs.lemma_oppositesideflip

Require Export GeoCoq.Elements.OriginalProofs.lemma_NCorder.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_oppositesideflip :
   ∀ A B P Q,
   TS P A B Q →
   TS P B A Q.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ r, (BetS P r Q ∧ Col A B r ∧ nCol A B P)) by (conclude_def TS );destruct Tf as [r];spliter.
assert (nCol B A P) by (forward_using lemma_NCorder).
assert (Col B A r) by (forward_using lemma_collinearorder).
assert (TS P B A Q) by (conclude_def TS ).
close.
Qed.

End Euclid.