proposition: Lemma proposition_30A : forall A B C D E F G H K, Par A B E F -> Par C D E F -> BetS G H K -> BetS A G B -> BetS E H F -> BetS C K D -> TS A G H F -> TS F H K C -> Par A B C D.
proof: Proof. intros. assert (Par E F C D) by (conclude lemma_parallelsymmetric). assert (neq G H) by (forward_using lemma_betweennotequal). assert (neq H G) by (conclude lemma_inequalitysymmetric). let Tf:=fresh in assert (Tf:exists P, (BetS H G P /\ Cong G P G H)) by (conclude lemma_extension);destruct Tf as [P];spliter. assert (BetS P G H) by (conclude axiom_betweennesssymmetry). assert (BetS P G K) by (conclude lemma_3_7b). let Tf:=fresh in assert (Tf:exists M, (BetS A M F /\ Col G H M /\ nCol G H A)) by (conclude_def TS );destruct Tf as [M];spliter. let Tf:=fresh in assert (Tf:exists N, (BetS F N C /\ Col H K N /\ nCol H K F)) by (conclude_def TS );destruct Tf as [N];spliter. assert (~ Meet C D E F) by (conclude_def Par ). assert (nCol G H A) by (conclude_def TS ). assert (neq A G) by (forward_using lemma_betweennotequal). assert (neq G A) by (conclude lemma_inequalitysymmetric). assert (neq G H) by (forward_using lemma_betweennotequal). assert (nCol A G H) by (forward_using lemma_NCorder). assert (CongA A G H G H F) by (conclude (proposition_29 A B E F P G H)). assert (eq A A) by (conclude cn_equalityreflexive). assert (Out G A A) by (conclude lemma_ray4). assert (Out G H K) by (conclude lemma_ray4). assert (CongA A G H A G H) by (conclude lemma_equalanglesreflexive). assert (CongA A G H A G K) by (conclude lemma_equalangleshelper). assert (CongA A G K A G H) by (conclude lemma_equalanglessymmetric). assert (CongA A G K G H F) by (conclude lemma_equalanglestransitive). assert (BetS C N F) by (conclude axiom_betweennesssymmetry). assert (eq H H) by (conclude cn_equalityreflexive). assert (nCol F H K) by (forward_using lemma_NCorder). assert (Col E H F) by (conclude_def Col ). assert (Col F H E) by (forward_using lemma_collinearorder). assert (Col F H H) by (conclude_def Col ). assert (neq E H) by (forward_using lemma_betweennotequal). assert (nCol E H K) by (conclude lemma_NChelper). assert (Col E H F) by (conclude_def Col ). assert (Col E H H) by (conclude_def Col ). assert (neq H F) by (forward_using lemma_betweennotequal). assert (neq F H) by (conclude lemma_inequalitysymmetric). assert (nCol F H K) by (conclude lemma_NChelper). assert (nCol H K F) by (forward_using lemma_NCorder). assert (eq H H) by (conclude cn_equalityreflexive). assert (Col H K H) by (conclude_def Col ). assert (Col K H N) by (forward_using lemma_collinearorder). assert (Col C K D) by (conclude_def Col ). assert (Col E H F) by (conclude_def Col ). assert (neq C K) by (forward_using lemma_betweennotequal). assert (neq C D) by (forward_using lemma_betweennotequal). assert (neq H F) by (forward_using lemma_betweennotequal). assert (neq E F) by (forward_using lemma_betweennotequal). assert (BetS K N H) by (conclude lemma_collinearbetween). assert (neq N H) by (forward_using lemma_betweennotequal). assert (neq H N) by (conclude lemma_inequalitysymmetric). assert (nCol H N F) by (conclude lemma_NChelper). assert (nCol F N H) by (forward_using lemma_NCorder). assert (BetS F N C) by (conclude axiom_betweennesssymmetry). assert (Col F N C) by (conclude_def Col ). assert (eq N N) by (conclude cn_equalityreflexive). assert (Col F N N) by (conclude_def Col ). assert (neq C N) by (forward_using lemma_betweennotequal). assert (nCol C N H) by (conclude lemma_NChelper). assert (nCol H N C) by (forward_using lemma_NCorder). assert (BetS H N K) by (conclude axiom_betweennesssymmetry). assert (Col H N K) by (conclude_def Col ). assert (Col H N H) by (conclude_def Col ). assert (neq H K) by (forward_using lemma_betweennotequal). assert (nCol H K C) by (conclude lemma_NChelper). assert (nCol H K E) by (forward_using lemma_NCorder). assert (OS E C H K) by (conclude_def OS ). assert (eq K K) by (conclude cn_equalityreflexive). assert (Col H K K) by (conclude_def Col ). assert (TS C H K D) by (conclude_def TS ). assert (TS E H K D) by (conclude lemma_planeseparation). assert (CongA G H F H K D) by (conclude proposition_29). assert (nCol C K H) by (forward_using lemma_NCorder). assert (Col C K D) by (conclude_def Col ). assert (eq K K) by (conclude cn_equalityreflexive). assert (Col C K K) by (conclude_def Col ). assert (neq K D) by (forward_using lemma_betweennotequal). assert (neq D K) by (conclude lemma_inequalitysymmetric). assert (nCol D K H) by (conclude lemma_NChelper). assert (nCol H K D) by (forward_using lemma_NCorder). assert (CongA H K D H K D) by (conclude lemma_equalanglesreflexive). assert (eq D D) by (conclude cn_equalityreflexive). assert (Out K D D) by (conclude lemma_ray4). assert (BetS K H G) by (conclude axiom_betweennesssymmetry). assert (neq K H) by (forward_using lemma_betweennotequal). assert (Out K H G) by (conclude lemma_ray4). assert (CongA H K D G K D) by (conclude lemma_equalangleshelper). assert (CongA G H F G K D) by (conclude lemma_equalanglestransitive). assert (CongA A G K G K D) by (conclude lemma_equalanglestransitive). assert (Col G H K) by (conclude_def Col ). assert (neq G H) by (forward_using lemma_betweennotequal). assert (Col H M K) by (conclude lemma_collinear4). assert (Col H K M) by (forward_using lemma_collinearorder). assert (Col H K G) by (forward_using lemma_collinearorder). assert (neq H K) by (forward_using lemma_betweennotequal). assert (Col K M G) by (conclude lemma_collinear4). assert (Col G K M) by (forward_using lemma_collinearorder). assert (Col H K G) by (forward_using lemma_collinearorder). assert (Col K N G) by (conclude lemma_collinear4). assert (Col G K N) by (forward_using lemma_collinearorder). assert (nCol A G K) by (conclude lemma_equalanglesNC). assert (nCol G K A) by (forward_using lemma_NCorder). assert (nCol H K C) by (forward_using lemma_NCorder). assert (Col H K G) by (forward_using lemma_collinearorder). assert (Col H K K) by (conclude_def Col ). assert (neq G K) by (forward_using lemma_betweennotequal). assert (nCol G K C) by (conclude lemma_NChelper). assert (OS A C G K) by (conclude_def OS ). assert (Col G K K) by (conclude_def Col ). assert (TS C G K D) by (conclude_def TS ). assert (TS A G K D) by (conclude lemma_planeseparation). assert (Par A B C D) by (conclude proposition_27). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric GeoCoq.Elements.OriginalProofs.proposition_29