proposition: Lemma lemma_parallelcollinear2 : forall A B C c d, TP A B c d -> BetS c C d -> TP A B C d.
proof: Proof. intros. assert (BetS d C c) by (conclude axiom_betweennesssymmetry). assert ((neq A B /\ neq c d /\ ~ Meet A B c d /\ OS c d A B)) by (conclude_def TP ). let Tf:=fresh in assert (Tf:exists p q r, (Col A B p /\ Col A B r /\ BetS c p q /\ BetS d r q /\ nCol A B c /\ nCol A B d)) by (conclude_def OS );destruct Tf as [p[q[r]]];spliter. assert (neq C d) by (forward_using lemma_betweennotequal). assert (BetS q p c) by (conclude axiom_betweennesssymmetry). assert (BetS q r d) by (conclude axiom_betweennesssymmetry). assert (BetS d C c) by (conclude axiom_betweennesssymmetry). assert (BetS q p c) by (conclude axiom_betweennesssymmetry). assert (Col q p c) by (conclude_def Col ). assert (~ eq p r). { intro. assert (Col q r d) by (conclude_def Col ). assert (Col q p c) by (conclude_def Col ). assert (Col q p d) by (conclude cn_equalitysub). assert (neq q p) by (forward_using lemma_betweennotequal). assert (Col p c d) by (conclude lemma_collinear4). assert (Col c d p) by (forward_using lemma_collinearorder). assert (Meet A B c d) by (conclude_def Meet ). contradict. } assert (nCol p r c) by (conclude lemma_NChelper). assert (nCol p r d) by (conclude lemma_NChelper). assert (nCol r d p) by (forward_using lemma_NCorder). assert (Col q r d) by (conclude_def Col ). assert (Col r d q) by (forward_using lemma_collinearorder). assert (eq d d) by (conclude cn_equalityreflexive). assert (Col r d d) by (conclude_def Col ). assert (neq q d) by (forward_using lemma_betweennotequal). assert (nCol q d p) by (conclude lemma_NChelper). assert (nCol q p d) by (forward_using lemma_NCorder). assert (Col q p c) by (conclude_def Col ). assert (eq c c) by (conclude cn_equalityreflexive). assert (~ eq c p). { intro. assert (eq p c) by (conclude lemma_equalitysymmetric). assert (Col p r c) by (conclude_def Col ). contradict. } assert (eq p p) by (conclude cn_equalityreflexive). assert (Col q p p) by (conclude_def Col ). assert (nCol c p d) by (conclude lemma_NChelper). assert (Col c p q) by (forward_using lemma_collinearorder). assert (eq c c) by (conclude cn_equalityreflexive). assert (Col c p c) by (conclude_def Col ). assert (neq q c) by (forward_using lemma_betweennotequal). assert (nCol q c d) by (conclude lemma_NChelper). assert (BetS q p c) by (conclude axiom_betweennesssymmetry). let Tf:=fresh in assert (Tf:exists E, (BetS q E C /\ BetS d E p)) by (conclude postulate_Pasch_inner);destruct Tf as [E];spliter. assert (BetS p E d) by (conclude axiom_betweennesssymmetry). assert (BetS q r d) by (conclude axiom_betweennesssymmetry). let Tf:=fresh in assert (Tf:exists F, (BetS q F E /\ BetS p F r)) by (conclude postulate_Pasch_inner);destruct Tf as [F];spliter. assert (Col p r F) by (conclude_def Col ). assert (Col B r p) by (conclude lemma_collinear4). assert (Col B A p) by (forward_using lemma_collinearorder). assert (Col B A r) by (forward_using lemma_collinearorder). assert (neq B A) by (conclude lemma_inequalitysymmetric). assert (Col B p r) by (forward_using lemma_collinearorder). assert (Col B p A) by (forward_using lemma_collinearorder). assert (~ Col A B C). { intro. assert (Col c C d) by (conclude_def Col ). assert (Col c d C) by (forward_using lemma_collinearorder). assert (Meet A B c d) by (conclude_def Meet ). contradict. } assert (BetS q F C) by (conclude lemma_3_6b). assert (BetS C F q) by (conclude axiom_betweennesssymmetry). assert (~ ~ OS C d A B). { intro. assert (~ neq B p). { intro. assert (Col p r A) by (conclude lemma_collinear4). assert (Col A p r) by (forward_using lemma_collinearorder). assert (Col A p B) by (forward_using lemma_collinearorder). assert (~ neq A p). { intro. assert (Col p r B) by (conclude lemma_collinear4). assert (Col A B F) by (conclude lemma_collinear5). assert (OS C d A B) by (conclude_def OS ). contradict. } assert (Col A r F) by (conclude cn_equalitysub). assert (Col r A F) by (forward_using lemma_collinearorder). assert (Col r A B) by (forward_using lemma_collinearorder). assert (~ eq r A). { intro. assert (eq r p) by (conclude cn_equalitysub). assert (neq p r) by (forward_using lemma_betweennotequal). assert (neq r p) by (conclude lemma_inequalitysymmetric). contradict. } assert (Col A F B) by (conclude lemma_collinear4). assert (Col A B F) by (forward_using lemma_collinearorder). assert (OS C d A B) by (conclude_def OS ). contradict. } assert (neq A p) by (conclude cn_equalitysub). assert (Col A p B) by (forward_using lemma_collinearorder). assert (eq A A) by (conclude cn_equalityreflexive). assert (Col B A A) by (conclude_def Col ). assert (Col B A p) by (forward_using lemma_collinearorder). assert (Col B A r) by (forward_using lemma_collinearorder). assert (Col A p r) by (conclude lemma_collinear5). assert (Col p B r) by (conclude lemma_collinear4). assert (Col p r B) by (forward_using lemma_collinearorder). assert (Col p r A) by (forward_using lemma_collinearorder). assert (Col A B F) by (conclude lemma_collinear5). assert (OS C d A B) by (conclude_def OS ). contradict. } assert (~ Meet A B C d). { intro. let Tf:=fresh in assert (Tf:exists R, (neq A B /\ neq C d /\ Col A B R /\ Col C d R)) by (conclude_def Meet );destruct Tf as [R];spliter. assert (Col c C d) by (conclude_def Col ). assert (Col C d c) by (forward_using lemma_collinearorder). assert (neq C d) by (forward_using lemma_betweennotequal). assert (Col d c R) by (conclude lemma_collinear4). assert (Col c d R) by (forward_using lemma_collinearorder). assert (Meet A B c d) by (conclude_def Meet ). contradict. } assert (TP A B C d) by (conclude_def TP ). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_NChelper GeoCoq.Elements.OriginalProofs.lemma_NCorder