Library GeoCoq.Elements.OriginalProofs.lemma_parallelflip

Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearorder.
Require Export GeoCoq.Elements.OriginalProofs.lemma_inequalitysymmetric.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_parallelflip :
   ∀ A B C D,
   Par A B C D →
   Par B A C D ∧ Par A B D C ∧ Par B A D C.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ M a b c d, (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 [M[a[b[c[d]]]]];spliter.
assert (Col B A a) by (forward_using lemma_collinearorder).
assert (Col B A b) by (forward_using lemma_collinearorder).
assert (Col D C c) by (forward_using lemma_collinearorder).
assert (Col D C d) by (forward_using lemma_collinearorder).
assert (neq B A) by (conclude lemma_inequalitysymmetric).
assert (neq D C) by (conclude lemma_inequalitysymmetric).
assert (BetS d M a) by (conclude axiom_betweennesssymmetry).
assert (BetS b M c) by (conclude axiom_betweennesssymmetry).
assert (neq d c) by (conclude lemma_inequalitysymmetric).
assert (neq b a) by (conclude lemma_inequalitysymmetric).
assert (¬ Meet A B D C).
 {
 intro.
 let Tf:=fresh in
 assert (Tf:∃ P, (neq A B ∧ neq D C ∧ Col A B P ∧ Col D C P)) by (conclude_def Meet );destruct Tf as [P];spliter.
 assert (Col C D P) by (forward_using lemma_collinearorder).
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (¬ Meet B A C D).
 {
 intro.
 let Tf:=fresh in
 assert (Tf:∃ P, (neq B A ∧ neq C D ∧ Col B A P ∧ Col C D P)) by (conclude_def Meet );destruct Tf as [P];spliter.
 assert (Col A B P) by (forward_using lemma_collinearorder).
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (¬ Meet B A D C).
 {
 intro.
 let Tf:=fresh in
 assert (Tf:∃ P, (neq B A ∧ neq D C ∧ Col B A P ∧ Col D C P)) by (conclude_def Meet );destruct Tf as [P];spliter.
 assert (Col A B P) by (forward_using lemma_collinearorder).
 assert (Col C D P) by (forward_using lemma_collinearorder).
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (Par B A C D) by (conclude_def Par ).
assert (Par A B D C) by (conclude_def Par ).
assert (Par B A D C) by (conclude_def Par ).
close.
Qed.

End Euclid.