proposition: Lemma lemma_crossbar : forall A B C E U V, Triangle A B C -> BetS A E C -> Out B A U -> Out B C V -> exists X, Out B E X /\ BetS U X V.
proof: Proof. intros. assert (nCol A B C) by (conclude_def Triangle ). assert (~ eq B E). { intro. assert (~ BetS A B C). { intro. assert (Col A B C) by (conclude_def Col ). contradict. } assert (BetS A B C) by (conclude cn_equalitysub). contradict. } assert (~ eq B A). { intro. assert (eq A B) by (conclude lemma_equalitysymmetric). assert (Col A B C) by (conclude_def Col ). contradict. } assert (~ eq B C). { intro. assert (Col A B C) by (conclude_def Col ). contradict. } assert (neq B U) by (conclude lemma_raystrict). assert (neq B V) by (conclude lemma_raystrict). let Tf:=fresh in assert (Tf:exists P, (BetS B A P /\ Cong A P B U)) by (conclude lemma_extension);destruct Tf as [P];spliter. let Tf:=fresh in assert (Tf:exists Q, (BetS B C Q /\ Cong C Q B V)) by (conclude lemma_extension);destruct Tf as [Q];spliter. assert (~ Col B Q A). { intro. assert (Col Q B A) by (forward_using lemma_collinearorder). assert (Col B C Q) by (conclude_def Col ). assert (Col Q B C) by (forward_using lemma_collinearorder). assert (neq B Q) by (forward_using lemma_betweennotequal). assert (neq Q B) by (conclude lemma_inequalitysymmetric). assert (Col B A C) by (conclude lemma_collinear4). assert (Col A B C) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists F, (BetS A F Q /\ BetS B E F)) by (conclude postulate_Pasch_outer);destruct Tf as [F];spliter. assert (BetS Q F A) by (conclude axiom_betweennesssymmetry). assert (~ Col B P Q). { intro. assert (Col P B Q) by (forward_using lemma_collinearorder). assert (Col B A P) by (conclude_def Col ). assert (Col P B A) by (forward_using lemma_collinearorder). assert (neq B P) by (forward_using lemma_betweennotequal). assert (neq P B) by (conclude lemma_inequalitysymmetric). assert (Col B Q A) by (conclude lemma_collinear4). contradict. } let Tf:=fresh in assert (Tf:exists W, (BetS Q W P /\ BetS B F W)) by (conclude postulate_Pasch_outer);destruct Tf as [W];spliter. assert (BetS B E W) by (conclude lemma_3_6b). let Tf:=fresh in assert (Tf:exists J, (BetS J B U /\ BetS J B A)) by (conclude_def Out );destruct Tf as [J];spliter. assert (Cong A P P A) by (conclude cn_equalityreverse). assert (Cong B U A P) by (conclude lemma_congruencesymmetric). assert (Cong B U P A) by (conclude lemma_congruencetransitive). assert (Cong P A B U) by (conclude lemma_congruencesymmetric). assert (BetS P A B) by (conclude axiom_betweennesssymmetry). assert (Lt B U P B) by (conclude_def Lt ). assert (Cong P B B P) by (conclude cn_equalityreverse). assert (Lt B U B P) by (conclude lemma_lessthancongruence). let Tf:=fresh in assert (Tf:exists S, (BetS B S P /\ Cong B S B U)) by (conclude_def Lt );destruct Tf as [S];spliter. assert (BetS J B P) by (conclude lemma_3_7b). assert (BetS J B S) by (conclude axiom_innertransitivity). assert (eq S U) by (conclude lemma_extensionunique). assert (BetS B U P) by (conclude cn_equalitysub). let Tf:=fresh in assert (Tf:exists K, (BetS K B V /\ BetS K B C)) by (conclude_def Out );destruct Tf as [K];spliter. assert (Cong B V C Q) by (conclude lemma_congruencesymmetric). assert (Cong C Q Q C) by (conclude cn_equalityreverse). assert (Cong B V Q C) by (conclude lemma_congruencetransitive). assert (Cong Q C B V) by (conclude lemma_congruencesymmetric). assert (BetS Q C B) by (conclude axiom_betweennesssymmetry). assert (Lt B V Q B) by (conclude_def Lt ). assert (Cong Q B B Q) by (conclude cn_equalityreverse). assert (Lt B V B Q) by (conclude lemma_lessthancongruence). let Tf:=fresh in assert (Tf:exists R, (BetS B R Q /\ Cong B R B V)) by (conclude_def Lt );destruct Tf as [R];spliter. assert (BetS K B Q) by (conclude lemma_3_7b). assert (BetS K B R) by (conclude axiom_innertransitivity). assert (eq R V) by (conclude lemma_extensionunique). assert (BetS B V Q) by (conclude cn_equalitysub). assert (~ Col Q P B). { intro. assert (Col B P Q) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists M, (BetS Q M U /\ BetS B M W)) by (conclude postulate_Pasch_inner);destruct Tf as [M];spliter. assert (BetS U M Q) by (conclude axiom_betweennesssymmetry). assert (~ Col U Q B). { intro. assert (Col B U P) by (conclude_def Col ). assert (Col B U Q) by (forward_using lemma_collinearorder). assert (neq B U) by (forward_using lemma_betweennotequal). assert (Col U B P) by (forward_using lemma_collinearorder). assert (Col U B Q) by (forward_using lemma_collinearorder). assert (neq U B) by (conclude lemma_inequalitysymmetric). assert (Col B P Q) by (conclude lemma_collinear4). assert (Col Q P B) by (forward_using lemma_collinearorder). contradict. } rename_H H;let Tf:=fresh in assert (Tf:exists H, (BetS U H V /\ BetS B H M)) by (conclude postulate_Pasch_inner);destruct Tf as [H];spliter. assert (~ eq E B). { intro. assert (eq B E) by (conclude lemma_equalitysymmetric). contradict. } let Tf:=fresh in assert (Tf:exists N, (BetS E B N /\ Cong B N B E)) by (conclude lemma_extension);destruct Tf as [N];spliter. assert (BetS N B E) by (conclude axiom_betweennesssymmetry). assert (BetS B H W) by (conclude lemma_3_6b). assert (BetS N B W) by (conclude lemma_3_7b). assert (BetS N B H) by (conclude axiom_innertransitivity). assert (Out B E H) by (conclude_def Out ). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_raystrict GeoCoq.Elements.OriginalProofs.lemma_collinear4 GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence