# 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.

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.