proposition: Lemma lemma_planeseparation : forall A B C D E, OS C D A B -> TS D A B E -> TS C A B E.
proof: Proof. intros. rename_H H; let Tf:=fresh in assert (Tf:exists G H Q, (Col A B G /\ Col A B H /\ BetS C G Q /\ BetS D H Q /\ nCol A B C /\ nCol A B D)) by (conclude_def OS );destruct Tf as [G[H[Q]]];spliter. assert (~ eq A B). { intro. assert (Col A B C) by (conclude_def Col ). contradict. } assert (neq B A) by (conclude lemma_inequalitysymmetric). let Tf:=fresh in assert (Tf:exists W, (BetS D W E /\ Col A B W /\ nCol A B D)) by (conclude_def TS );destruct Tf as [W];spliter. assert (neq C G) by (forward_using lemma_betweennotequal). assert (neq G C) by (conclude lemma_inequalitysymmetric). assert (neq G Q) by (forward_using lemma_betweennotequal). assert (TS C A B E). by cases on (Col C Q D \/ nCol C Q D). { assert (~ ~ TS C A B E). { intro. assert (Col Q C D) by (forward_using lemma_collinearorder). assert (~ neq G H). { intro. assert (neq D Q) by (forward_using lemma_betweennotequal). assert (Col C G Q) by (conclude_def Col ). assert (Col D H Q) by (conclude_def Col ). assert (Col Q C G) by (forward_using lemma_collinearorder). assert (neq C Q) by (forward_using lemma_betweennotequal). assert (neq Q C) by (conclude lemma_inequalitysymmetric). assert (Col C G D) by (conclude lemma_collinear4). assert (Col D Q H) by (forward_using lemma_collinearorder). assert (Col D Q C) by (forward_using lemma_collinearorder). assert (Col Q H C) by (conclude lemma_collinear4). assert (Col Q C H) by (forward_using lemma_collinearorder). assert (~ Col C A B). { intro. assert (Col A B C) by (forward_using lemma_collinearorder). contradict. } assert (~ (Col C A B /\ Col D A B)). { intro. contradict. } assert (Col G A B) by (forward_using lemma_collinearorder). assert (Col H A B) by (forward_using lemma_collinearorder). assert (Col G C D) by (forward_using lemma_collinearorder). assert (Col Q D H) by (forward_using lemma_collinearorder). assert (Col Q D C) by (forward_using lemma_collinearorder). assert (neq D Q) by (forward_using lemma_betweennotequal). assert (neq Q D) by (conclude lemma_inequalitysymmetric). assert (Col D C H) by (conclude lemma_collinear4). assert (Col H C D) by (forward_using lemma_collinearorder). assert (~ eq C D). { intro. assert (TS C A B E) by (conclude cn_equalitysub). contradict. } assert (eq G H) by (conclude lemma_twolines2). contradict. } assert (BetS Q G C) by (conclude axiom_betweennesssymmetry). assert (BetS Q H D) by (conclude axiom_betweennesssymmetry). assert (BetS Q G D) by (conclude cn_equalitysub). assert (~ nCol E D G). { intro. assert (~ nCol E C G). { intro. assert (~ BetS G C D). { intro. assert (TS C A B E) by (conclude lemma_9_5b). contradict. } assert (~ BetS G D C). { intro. assert (~ Col G C E). { intro. assert (Col E C G) by (forward_using lemma_collinearorder). contradict. } assert (TS C A B E) by (conclude lemma_9_5a). contradict. } assert (eq C D) by (conclude lemma_outerconnectivity). assert (TS C A B E) by (conclude cn_equalitysub). contradict. } assert (Col C G E) by (forward_using lemma_collinearorder). assert (Col Q G D) by (conclude_def Col ). assert (Col Q G C) by (conclude_def Col ). assert (neq Q G) by (forward_using lemma_betweennotequal). assert (Col G D C) by (conclude lemma_collinear4). assert (Col C G D) by (forward_using lemma_collinearorder). assert (Col C G E) by (forward_using lemma_collinearorder). assert (neq G C) by (forward_using lemma_betweennotequal). assert (neq C G) by (conclude lemma_inequalitysymmetric). assert (Col G D E) by (conclude lemma_collinear4). assert (Col E D G) by (forward_using lemma_collinearorder). contradict. } assert (Col D G E) by (forward_using lemma_collinearorder). assert (Col D H Q) by (conclude_def Col ). assert (Col D G Q) by (conclude cn_equalitysub). assert (neq G D) by (forward_using lemma_betweennotequal). assert (neq D G) by (conclude lemma_inequalitysymmetric). assert (Col G E Q) by (conclude lemma_collinear4). assert (Col G E D) by (forward_using lemma_collinearorder). assert (Col D W E) by (conclude_def Col ). assert (Col D E W) by (forward_using lemma_collinearorder). assert (Col D E G) by (forward_using lemma_collinearorder). assert (neq D E) by (forward_using lemma_betweennotequal). assert (Col E W G) by (conclude lemma_collinear4). assert (Col E W D) by (forward_using lemma_collinearorder). assert (neq W E) by (forward_using lemma_betweennotequal). assert (neq E W) by (conclude lemma_inequalitysymmetric). assert (Col W G D) by (conclude lemma_collinear4). assert (Col B W G) by (conclude lemma_collinear4). assert (Col W G B) by (forward_using lemma_collinearorder). assert (Col B A W) by (forward_using lemma_collinearorder). assert (Col B A G) by (forward_using lemma_collinearorder). assert (Col A W G) by (conclude lemma_collinear4). assert (Col W G A) by (forward_using lemma_collinearorder). assert (~ neq W G). { intro. assert (Col G D B) by (conclude lemma_collinear4). assert (Col G B D) by (forward_using lemma_collinearorder). assert (Col G B A) by (forward_using lemma_collinearorder). assert (~ neq G B). { intro. assert (Col B D A) by (conclude lemma_collinear4). assert (Col A B D) by (forward_using lemma_collinearorder). contradict. } assert (Col G D A) by (conclude lemma_collinear4). assert (Col G A D) by (forward_using lemma_collinearorder). assert (Col G A B) by (forward_using lemma_collinearorder). assert (~ neq G A). { intro. assert (Col A D B) by (conclude lemma_collinear4). assert (Col A B D) by (forward_using lemma_collinearorder). contradict. } assert (eq A G) by (conclude lemma_equalitysymmetric). assert (eq B G) by (conclude lemma_equalitysymmetric). assert (eq A B) by (conclude cn_equalitytransitive). contradict. } assert (BetS D G E) by (conclude cn_equalitysub). assert (BetS E G D) by (conclude axiom_betweennesssymmetry). assert (~ BetS G D C). { intro. assert (BetS E G C) by (conclude lemma_3_7b). assert (BetS C G E) by (conclude axiom_betweennesssymmetry). assert (TS C A B E) by (conclude_def TS ). contradict. } assert (~ BetS G C D). { intro. assert (BetS E G C) by (conclude axiom_innertransitivity). assert (BetS C G E) by (conclude axiom_betweennesssymmetry). assert (TS C A B E) by (conclude_def TS ). contradict. } assert (~ BetS C G D). { intro. assert (BetS D G Q) by (conclude cn_equalitysub). assert (BetS Q G C) by (conclude axiom_betweennesssymmetry). assert (BetS D G C) by (conclude axiom_betweennesssymmetry). assert (neq G D) by (forward_using lemma_betweennotequal). assert (~ BetS G D C). { intro. assert (BetS C D G) by (conclude axiom_betweennesssymmetry). assert (BetS C G C) by (conclude lemma_3_7a). assert (neq C C) by (forward_using lemma_betweennotequal). assert (eq C C) by (conclude cn_equalityreflexive). contradict. } assert (~ BetS G C D). { intro. assert (BetS D G C) by (conclude axiom_betweennesssymmetry). assert (BetS C G C) by (conclude axiom_innertransitivity). assert (neq C C) by (forward_using lemma_betweennotequal). assert (eq C C) by (conclude cn_equalityreflexive). contradict. } assert (BetS Q G D) by (conclude axiom_betweennesssymmetry). assert (BetS Q G C) by (conclude axiom_betweennesssymmetry). assert (eq C D) by (conclude lemma_outerconnectivity). assert (TS C A B E) by (conclude cn_equalitysub). contradict. } assert (Col Q C D) by (forward_using lemma_collinearorder). assert (Col Q G C) by (conclude_def Col ). assert (Col Q C G) by (forward_using lemma_collinearorder). assert (neq Q C) by (forward_using lemma_betweennotequal). assert (Col C D G) by (conclude lemma_collinear4). assert (Col G C D) by (forward_using lemma_collinearorder). assert ((eq G C \/ eq G D \/ eq C D \/ BetS C G D \/ BetS G C D \/ BetS G D C)) by (conclude_def Col ). assert (TS C A B E). by cases on (eq G C \/ eq G D \/ eq C D \/ BetS C G D \/ BetS G C D \/ BetS G D C). { assert (~ ~ TS C A B E). { intro. assert (Col A B C) by (conclude cn_equalitysub). contradict. } close. } { assert (~ ~ TS C A B E). { intro. assert (Col A B D) by (conclude cn_equalitysub). contradict. } close. } { assert (~ ~ TS C A B E). { intro. assert (TS C A B E) by (conclude cn_equalitysub). contradict. } close. } { assert (~ ~ TS C A B E). { intro. contradict. } close. } { assert (~ ~ TS C A B E). { intro. contradict. } close. } { assert (~ ~ TS C A B E). { intro. contradict. } close. } contradict. } close. } { assert (~ ~ TS C A B E). { intro. assert (~ Col Q C D). { intro. assert (Col C Q D) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists F, (BetS C F H /\ BetS D F G)) by (conclude postulate_Pasch_inner);destruct Tf as [F];spliter. assert (BetS H F C) by (conclude axiom_betweennesssymmetry). assert (BetS G F D) by (conclude axiom_betweennesssymmetry). assert (~ Col E D G). { intro. assert (~ neq W G). { intro. assert (Col D E G) by (forward_using lemma_collinearorder). assert (Col D W E) by (conclude_def Col ). assert (Col B G W) by (conclude lemma_collinear4). assert (Col W B G) by (forward_using lemma_collinearorder). assert (Col W B A) by (forward_using lemma_collinearorder). assert (Col B G A). by cases on (eq W B \/ neq W B). { assert (Col B A G) by (forward_using lemma_collinearorder). assert (Col B A W) by (forward_using lemma_collinearorder). assert (Col A G W) by (conclude lemma_collinear4). assert (Col A G B) by (conclude cn_equalitysub). assert (Col B G A) by (forward_using lemma_collinearorder). close. } { assert (Col B G A) by (conclude lemma_collinear4). close. } assert (Col G A B) by (forward_using lemma_collinearorder). assert (Col E D G) by (forward_using lemma_collinearorder). assert (Col E D W) by (forward_using lemma_collinearorder). assert (neq D E) by (forward_using lemma_betweennotequal). assert (neq E D) by (conclude lemma_inequalitysymmetric). assert (Col D G W) by (conclude lemma_collinear4). assert (Col G W D) by (forward_using lemma_collinearorder). assert (eq B B) by (conclude cn_equalityreflexive). assert (Col A B B) by (conclude_def Col ). assert (neq G W) by (conclude lemma_inequalitysymmetric). assert (Col G W B) by (conclude lemma_collinear5). assert (Col W D B) by (conclude lemma_collinear4). assert (Col W B D) by (forward_using lemma_collinearorder). assert (Col B A W) by (forward_using lemma_collinearorder). assert (Col B A G) by (forward_using lemma_collinearorder). assert (eq A A) by (conclude cn_equalityreflexive). assert (Col B A A) by (conclude_def Col ). assert (Col G W A) by (conclude lemma_collinear5). assert (Col W D A) by (conclude lemma_collinear4). assert (neq D W) by (forward_using lemma_betweennotequal). assert (neq W D) by (conclude lemma_inequalitysymmetric). assert (Col D B A) by (conclude lemma_collinear4). assert (Col A B D) by (forward_using lemma_collinearorder). contradict. } assert (BetS D G E) by (conclude cn_equalitysub). assert (BetS E G D) by (conclude axiom_betweennesssymmetry). assert (BetS G F D) by (conclude axiom_betweennesssymmetry). assert (BetS E G F) by (conclude axiom_innertransitivity). assert (~ Col H C E). { intro. assert (Col H E C) by (forward_using lemma_collinearorder). assert (Col E H C) by (forward_using lemma_collinearorder). assert (Col C F H) by (conclude_def Col ). assert (Col H C F) by (forward_using lemma_collinearorder). assert (Col H C E) by (forward_using lemma_collinearorder). assert (neq C H) by (forward_using lemma_betweennotequal). assert (neq H C) by (conclude lemma_inequalitysymmetric). assert (Col C F E) by (conclude lemma_collinear4). assert (Col E F C) by (forward_using lemma_collinearorder). assert (BetS D F E) by (conclude lemma_3_6b). assert (Col D F E) by (conclude_def Col ). assert (Col E F D) by (forward_using lemma_collinearorder). assert (neq F E) by (forward_using lemma_betweennotequal). assert (neq E F) by (conclude lemma_inequalitysymmetric). assert (Col F C D) by (conclude lemma_collinear4). assert (Col F D C) by (forward_using lemma_collinearorder). assert (Col D F G) by (conclude_def Col ). assert (Col F D G) by (forward_using lemma_collinearorder). assert (neq D F) by (forward_using lemma_betweennotequal). assert (neq F D) by (conclude lemma_inequalitysymmetric). assert (Col D C G) by (conclude lemma_collinear4). assert (Col G C D) by (forward_using lemma_collinearorder). assert (Col C G Q) by (conclude_def Col ). assert (Col G C Q) by (forward_using lemma_collinearorder). assert (neq C G) by (forward_using lemma_betweennotequal). assert (neq G C) by (conclude lemma_inequalitysymmetric). assert (Col C D Q) by (conclude lemma_collinear4). assert (Col Q C D) by (forward_using lemma_collinearorder). contradict. } let Tf:=fresh in assert (Tf:exists M, (BetS E M C /\ BetS H G M)) by (conclude postulate_Pasch_outer);destruct Tf as [M];spliter. assert (Col H G M) by (conclude_def Col ). assert (Col B H G) by (conclude lemma_collinear4). assert (Col H G B) by (forward_using lemma_collinearorder). assert (neq H G) by (forward_using lemma_betweennotequal). assert (Col G M B) by (conclude lemma_collinear4). assert (Col G B M) by (forward_using lemma_collinearorder). assert (Col G B A) by (forward_using lemma_collinearorder). assert (Col A B M). by cases on (eq B G \/ neq B G). { assert (Col B A H) by (forward_using lemma_collinearorder). assert (Col B A G) by (forward_using lemma_collinearorder). assert (Col A H G) by (conclude lemma_collinear4). assert (Col H G A) by (forward_using lemma_collinearorder). assert (Col G A M) by (conclude lemma_collinear4). assert (Col B A M) by (conclude cn_equalitysub). assert (Col A B M) by (forward_using lemma_collinearorder). close. } { assert (neq G B) by (conclude lemma_inequalitysymmetric). assert (Col B M A) by (conclude lemma_collinear4). assert (Col A B M) by (forward_using lemma_collinearorder). close. } assert (BetS C M E) by (conclude axiom_betweennesssymmetry). assert (TS C A B E) by (conclude_def TS ). contradict. } assert (TS F A B E) by (conclude lemma_9_5b). assert (~ eq G H). { intro. assert (Col D H Q) by (conclude_def Col ). assert (Col C G Q) by (conclude_def Col ). assert (Col Q H D) by (forward_using lemma_collinearorder). assert (Col Q G C) by (forward_using lemma_collinearorder). assert (Col Q H C) by (conclude cn_equalitysub). assert (neq H Q) by (forward_using lemma_betweennotequal). assert (neq Q H) by (conclude lemma_inequalitysymmetric). assert (Col H D C) by (conclude lemma_collinear4). assert (Col H D Q) by (forward_using lemma_collinearorder). assert (neq D H) by (forward_using lemma_betweennotequal). assert (neq H D) by (conclude lemma_inequalitysymmetric). assert (Col D C Q) by (conclude lemma_collinear4). assert (Col C Q D) by (forward_using lemma_collinearorder). contradict. } assert (~ Col H C E). { intro. let Tf:=fresh in assert (Tf:exists K, (BetS G K H /\ Cong K G K H)) by (conclude proposition_10);destruct Tf as [K];spliter. assert (~ Col G D E). { intro. assert (Col E D G) by (forward_using lemma_collinearorder). contradict. } assert (BetS E W D) by (conclude axiom_betweennesssymmetry). assert (BetS G F D) by (conclude axiom_betweennesssymmetry). let Tf:=fresh in assert (Tf:exists J, (BetS E J F /\ BetS G J W)) by (conclude postulate_Pasch_inner);destruct Tf as [J];spliter. assert (Col G J W) by (conclude_def Col ). assert (Col E F J) by (conclude_def Col ). assert (Col E C H) by (forward_using lemma_collinearorder). assert (Col F E J) by (forward_using lemma_collinearorder). assert (Col C F H) by (conclude_def Col ). assert (Col C H F) by (forward_using lemma_collinearorder). assert (Col C H E) by (forward_using lemma_collinearorder). assert (neq C H) by (forward_using lemma_betweennotequal). assert (Col H F E) by (conclude lemma_collinear4). assert (Col F E H) by (forward_using lemma_collinearorder). assert (neq E F) by (forward_using lemma_betweennotequal). assert (neq F E) by (conclude lemma_inequalitysymmetric). assert (Col E J H) by (conclude lemma_collinear4). assert (Col E H J) by (forward_using lemma_collinearorder). assert (Col G W J) by (forward_using lemma_collinearorder). assert (eq H H) by (conclude cn_equalityreflexive). assert (Col E H H) by (conclude_def Col ). assert (Col G H W) by (conclude lemma_collinear5). assert (Col G W H) by (forward_using lemma_collinearorder). assert (~ eq G W). { intro. assert (Col D W E) by (conclude_def Col ). assert (Col D G E) by (conclude cn_equalitysub). assert (Col E D G) by (forward_using lemma_collinearorder). contradict. } assert (Col E J H) by (conclude lemma_collinear4). assert (Col W J H) by (conclude lemma_collinear4). assert (Col J H E) by (forward_using lemma_collinearorder). assert (Col J H W) by (forward_using lemma_collinearorder). assert (~ eq H W). { intro. assert (Col D W E) by (conclude_def Col ). assert (Col D H E) by (conclude cn_equalitysub). assert (Col D H Q) by (conclude_def Col ). assert (neq D H) by (forward_using lemma_betweennotequal). assert (Col H E Q) by (conclude lemma_collinear4). assert (Col H E C) by (forward_using lemma_collinearorder). assert (neq W E) by (forward_using lemma_betweennotequal). assert (neq H E) by (conclude cn_equalitysub). assert (Col E Q C) by (conclude lemma_collinear4). assert (Col E C Q) by (forward_using lemma_collinearorder). assert (Col E C H) by (forward_using lemma_collinearorder). assert (~ neq E C). { intro. assert (Col C Q H) by (conclude lemma_collinear4). assert (Col H Q C) by (forward_using lemma_collinearorder). assert (Col H Q D) by (forward_using lemma_collinearorder). assert (neq H Q) by (forward_using lemma_betweennotequal). assert (Col Q C D) by (conclude lemma_collinear4). contradict. } assert (Col E W D) by (forward_using lemma_collinearorder). assert (Col C W D) by (conclude cn_equalitysub). assert (Col C H D) by (conclude cn_equalitysub). assert (Col D H C) by (forward_using lemma_collinearorder). assert (Col D H Q) by (conclude_def Col ). assert (neq D H) by (forward_using lemma_betweennotequal). assert (Col H C Q) by (conclude lemma_collinear4). assert (Col H Q C) by (forward_using lemma_collinearorder). assert (Col H Q D) by (forward_using lemma_collinearorder). assert (neq H Q) by (forward_using lemma_betweennotequal). assert (Col Q C D) by (conclude lemma_collinear4). assert (Col C Q D) by (forward_using lemma_collinearorder). contradict. } assert (~ neq J H). { intro. assert (Col H E W) by (conclude lemma_collinear4). assert (Col H W E) by (forward_using lemma_collinearorder). assert (Col H W G) by (forward_using lemma_collinearorder). assert (Col W E G) by (conclude lemma_collinear4). assert (Col D W E) by (conclude_def Col ). assert (Col W E D) by (forward_using lemma_collinearorder). assert (neq W E) by (forward_using lemma_betweennotequal). assert (Col E G D) by (conclude lemma_collinear4). assert (Col E D G) by (forward_using lemma_collinearorder). contradict. } assert (BetS E H F) by (conclude cn_equalitysub). assert (BetS F H E) by (conclude axiom_betweennesssymmetry). assert (BetS C H E) by (conclude lemma_3_7a). assert (TS C A B E) by (conclude_def TS ). contradict. } assert (TS C A B E) by (conclude lemma_9_5a). contradict. } close. } close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_twolines2 GeoCoq.Elements.OriginalProofs.lemma_9_5b GeoCoq.Elements.OriginalProofs.lemma_9_5a GeoCoq.Elements.OriginalProofs.lemma_collinear5 GeoCoq.Elements.OriginalProofs.proposition_10