proposition: Lemma lemma_parallelflip : forall 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: Proof. intros. let Tf:=fresh in assert (Tf:exists 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:exists 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:exists 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:exists 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.
imports: GeoCoq.Elements.OriginalProofs.lemma_collinearorder GeoCoq.Elements.OriginalProofs.lemma_inequalitysymmetric