Library GeoCoq.Elements.OriginalProofs.lemma_samesidecollinear

Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.
Require Export GeoCoq.Elements.OriginalProofs.lemma_NChelper.

Section Euclid.

Context `{Ax1:euclidean_neutral}.

Lemma lemma_samesidecollinear :
   ∀ A B C P Q,
   OS P Q A B → Col A B C → neq A C →
   OS P Q A C.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ p q r, (Col A B p ∧ Col A B q ∧ BetS P p r ∧ BetS Q q r ∧ nCol A B P ∧ nCol A B Q)) by (conclude_def OS );destruct Tf as [p[q[r]]];spliter.
assert (neq A B) by (forward_using lemma_NCdistinct).
assert (eq A A) by (conclude cn_equalityreflexive).
assert (Col A B A) by (conclude_def Col ).
assert (nCol A C P) by (conclude lemma_NChelper).
assert (nCol A C Q) by (conclude lemma_NChelper).
assert (Col B A p) by (forward_using lemma_collinearorder).
assert (Col B A C) by (forward_using lemma_collinearorder).
assert (neq B A) by (conclude lemma_inequalitysymmetric).
assert (Col A C p) by (conclude lemma_collinear4).
assert (Col B A q) by (forward_using lemma_collinearorder).
assert (Col A C q) by (conclude lemma_collinear4).
assert (OS P Q A C) by (conclude_def OS ).
close.
Qed.

End Euclid.