proposition: Lemma lemma_angleordertransitive : forall A B C D E F P Q R, LtA A B C D E F -> LtA D E F P Q R -> LtA A B C P Q R.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists U V W, (BetS U W V /\ Out Q P U /\ Out Q R V /\ CongA D E F P Q W)) by (conclude_def LtA );destruct Tf as [U[V[W]]];spliter. assert (CongA P Q W D E F) by (conclude lemma_equalanglessymmetric). assert (neq D E) by (forward_using lemma_angledistinct). assert (neq E D) by (conclude lemma_inequalitysymmetric). assert (neq E F) by (forward_using lemma_angledistinct). assert (neq Q U) by (conclude lemma_raystrict). let Tf:=fresh in assert (Tf:exists G, (Out E D G /\ Cong E G Q U)) by (conclude lemma_layoff);destruct Tf as [G];spliter. assert (neq Q W) by (forward_using lemma_angledistinct). let Tf:=fresh in assert (Tf:exists J, (Out E F J /\ Cong E J Q W)) by (conclude lemma_layoff);destruct Tf as [J];spliter. assert (nCol D E F) by (conclude lemma_equalanglesNC). assert (CongA D E F D E F) by (conclude lemma_equalanglesreflexive). assert (CongA D E F G E J) by (conclude lemma_equalangleshelper). assert (CongA G E J D E F) by (conclude lemma_equalanglessymmetric). assert (CongA G E J P Q W) by (conclude lemma_equalanglestransitive). assert (eq W W) by (conclude cn_equalityreflexive). assert (Out Q W W) by (conclude lemma_ray4). assert (CongA G E J U Q W) by (conclude lemma_equalangleshelper). assert (nCol G E J) by (conclude lemma_equalanglesNC). assert (nCol U Q W) by (conclude lemma_equalanglesNC). assert (Triangle G E J) by (conclude_def Triangle ). assert (Triangle U Q W) by (conclude_def Triangle ). assert (Cong G J U W) by (conclude proposition_04). assert (eq W W) by (conclude cn_equalityreflexive). assert (CongA D E F U Q W) by (conclude lemma_equalangleshelper). assert (CongA U Q W D E F) by (conclude lemma_equalanglessymmetric). assert (CongA D E F U Q W) by (conclude lemma_equalangleshelper). assert (LtA A B C U Q W) by (conclude lemma_angleorderrespectscongruence). rename_H H; let Tf:=fresh in assert (Tf:exists H S T, (BetS S H T /\ Out Q U S /\ Out Q W T /\ CongA A B C U Q H)) by (conclude_def LtA );destruct Tf as [H[S[T]]];spliter. assert (Out Q U P) by (conclude lemma_ray5). assert (neq Q H) by (forward_using lemma_angledistinct). assert (eq H H) by (conclude cn_equalityreflexive). assert (Out Q H H) by (conclude lemma_ray4). assert (CongA A B C P Q H) by (conclude lemma_equalangleshelper). assert (CongA D E F P Q T) by (conclude lemma_equalangleshelper). assert (nCol P Q T) by (conclude lemma_equalanglesNC). assert (Triangle P Q T) by (conclude_def Triangle ). assert (neq Q P) by (conclude lemma_ray2). assert (Out Q T W) by (conclude lemma_ray5). assert (~ Col S Q T). { intro. assert (Col Q U S) by (conclude lemma_rayimpliescollinear). assert (Col U Q S) by (forward_using lemma_collinearorder). assert (Col Q P U) by (conclude lemma_rayimpliescollinear). assert (Col U Q P) by (forward_using lemma_collinearorder). assert (neq Q U) by (conclude lemma_raystrict). assert (neq U Q) by (conclude lemma_inequalitysymmetric). assert (Col Q S P) by (conclude lemma_collinear4). assert (Col S Q P) by (forward_using lemma_collinearorder). assert (neq Q S) by (conclude lemma_raystrict). assert (neq S Q) by (conclude lemma_inequalitysymmetric). assert (Col Q T P) by (conclude lemma_collinear4). assert (Col P Q T) by (forward_using lemma_collinearorder). contradict. } assert (Triangle S Q T) by (conclude_def Triangle ). assert (Out Q S U) by (conclude lemma_ray5). let Tf:=fresh in assert (Tf:exists K, (Out Q H K /\ BetS U K W)) by (conclude lemma_crossbar);destruct Tf as [K];spliter. assert (BetS U K V) by (conclude lemma_3_6b). assert (eq P P) by (conclude cn_equalityreflexive). assert (Out Q P P) by (conclude lemma_ray4). assert (CongA A B C P Q K) by (conclude lemma_equalangleshelper). assert (LtA A B C P Q R) by (conclude_def LtA ). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_equalanglesreflexive GeoCoq.Elements.OriginalProofs.lemma_angleorderrespectscongruence GeoCoq.Elements.OriginalProofs.lemma_crossbar