proposition: Lemma proposition_33 : forall A B C D M, Par A B C D -> Cong A B C D -> BetS A M D -> BetS B M C -> Par A C B D /\ Cong A C B D.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists a b c d m, (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 [a[b[c[d[m]]]]];spliter. assert (Col B M C) by (conclude_def Col ). assert (Col B C M) by (forward_using lemma_collinearorder). assert (~ Col B C A). { intro. assert (Col A B C) by (forward_using lemma_collinearorder). assert (eq C C) by (conclude cn_equalityreflexive). assert (Col C D C) by (conclude_def Col ). assert (Meet A B C D) by (conclude_def Meet ). contradict. } assert (TS A B C D) by (conclude_def TS ). assert (CongA A B C B C D) by (conclude proposition_29B). assert (~ Col B C D). { intro. assert (Col C D B) by (forward_using lemma_collinearorder). assert (eq B B) by (conclude cn_equalityreflexive). assert (Col A B B) by (conclude_def Col ). assert (Meet A B C D) by (conclude_def Meet ). contradict. } assert (CongA B C D D C B) by (conclude lemma_ABCequalsCBA). assert (CongA A B C D C B) by (conclude lemma_equalanglestransitive). assert (Cong B C B C) by (conclude cn_congruencereflexive). assert (nCol A B C) by (assert (nCol B C A) by auto; (forward_using lemma_NCorder)). assert (Cong B A C D) by (forward_using lemma_congruenceflip). assert (Cong B C C B) by (forward_using lemma_congruenceflip). assert ((Cong A C D B /\ CongA B A C C D B /\ CongA B C A C B D)) by (conclude proposition_04). assert (nCol A C B) by (forward_using lemma_NCorder). assert (CongA A C B B C A) by (conclude lemma_ABCequalsCBA). assert (CongA A C B C B D) by (conclude lemma_equalanglestransitive). assert (Cong A C B D) by (forward_using lemma_congruenceflip). assert (Col C B M) by (forward_using lemma_collinearorder). assert (nCol C B A) by (forward_using lemma_NCorder). assert (TS A C B D) by (conclude_def TS ). assert (Par A C B D) by (conclude proposition_27B). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.proposition_29B GeoCoq.Elements.OriginalProofs.proposition_27B