proposition: Lemma lemma_angleorderrespectscongruence : forall A B C D E F P Q R, LtA A B C D E F -> CongA P Q R D E F -> LtA A B C P Q R.
proof: Proof. intros. rename_H H;let Tf:=fresh in assert (Tf:exists G H J, (BetS G H J /\ Out E D G /\ Out E F J /\ CongA A B C D E H)) by (conclude_def LtA );destruct Tf as [G[H[J]]];spliter. assert ((neq P Q /\ neq Q R /\ neq P R /\ neq D E /\ neq E F /\ neq D F)) by (forward_using lemma_angledistinct). assert (neq Q P) by (conclude lemma_inequalitysymmetric). assert ((neq A B /\ neq B C /\ neq A C /\ neq D E /\ neq E H /\ neq D H)) by (forward_using lemma_angledistinct). assert (neq E G) by (conclude lemma_raystrict). let Tf:=fresh in assert (Tf:exists U, (Out Q P U /\ Cong Q U E G)) by (conclude lemma_layoff);destruct Tf as [U];spliter. assert (neq E J) by (conclude lemma_raystrict). let Tf:=fresh in assert (Tf:exists V, (Out Q R V /\ Cong Q V E J)) by (conclude lemma_layoff);destruct Tf as [V];spliter. assert (Cong G H G H) by (conclude cn_congruencereflexive). assert (Lt G H G J) by (conclude_def Lt ). assert (CongA D E F P Q R) by (conclude lemma_equalanglessymmetric). assert (CongA D E F U Q V) by (conclude lemma_equalangleshelper). assert (CongA U Q V D E F) by (conclude lemma_equalanglessymmetric). assert (CongA U Q V G E J) by (conclude lemma_equalangleshelper). assert (CongA G E J U Q V) by (conclude lemma_equalanglessymmetric). assert (Cong E G Q U) by (conclude lemma_congruencesymmetric). assert (Cong E J Q V) by (conclude lemma_congruencesymmetric). assert ((Cong G J U V /\ CongA E G J Q U V /\ CongA E J G Q V U)) by (conclude proposition_04). assert (Cong U V G J) by (conclude lemma_congruencesymmetric). assert (neq G J) by (forward_using lemma_betweennotequal). let Tf:=fresh in assert (Tf:exists W, (BetS U W V /\ Cong U W G H)) by (conclude proposition_03);destruct Tf as [W];spliter. assert (eq H H) by (conclude cn_equalityreflexive). assert (Out E H H) by (conclude lemma_ray4). assert (CongA A B C G E H) by (conclude lemma_equalangleshelper). assert (nCol G E H) by (conclude lemma_equalanglesNC). assert (nCol G H E) by (forward_using lemma_NCorder). assert (neq U V) by (forward_using lemma_betweennotequal). assert (Out U V W) by (conclude lemma_ray4). assert (eq Q Q) by (conclude cn_equalityreflexive). assert (eq E E) by (conclude cn_equalityreflexive). assert (nCol U Q V) by (conclude lemma_equalanglesNC). assert (neq U Q) by (forward_using lemma_NCdistinct). assert (Out U Q Q) by (conclude lemma_ray4). assert (~ eq G E). { intro. assert (Col G H E) by (conclude_def Col ). contradict. } assert (Out G E E) by (conclude lemma_ray4). assert (CongA E G J Q U W) by (conclude lemma_equalangleshelper). assert (CongA Q U W E G J) by (conclude lemma_equalanglessymmetric). assert (Out G J H) by (conclude lemma_ray4). assert (CongA Q U W E G H) by (conclude lemma_equalangleshelper). assert (CongA E G H Q U W) by (conclude lemma_equalanglessymmetric). assert (nCol Q U W) by (conclude lemma_equalanglesNC). assert (nCol U W Q) by (forward_using lemma_NCorder). assert (nCol H G E) by (forward_using lemma_NCorder). assert (~ Col W U Q). { intro. assert (Col U W Q) by (forward_using lemma_collinearorder). contradict. } assert (Cong G H U W) by (conclude lemma_congruencesymmetric). assert (Cong G E U Q) by (forward_using lemma_congruenceflip). assert (CongA E G H Q U W) by (conclude lemma_equalanglessymmetric). assert (CongA Q U W W U Q) by (conclude lemma_ABCequalsCBA). assert (CongA E G H W U Q) by (conclude lemma_equalanglestransitive). assert (CongA H G E E G H) by (conclude lemma_ABCequalsCBA). assert (CongA H G E W U Q) by (conclude lemma_equalanglestransitive). assert ((Cong H E W Q /\ CongA G H E U W Q /\ CongA G E H U Q W)) by (conclude proposition_04). assert (CongA A B C U Q W) by (conclude lemma_equalanglestransitive). assert (eq W W) by (conclude cn_equalityreflexive). assert (~ eq Q W). { intro. assert (Col Q U W) by (conclude_def Col ). assert (Col W U Q) by (forward_using lemma_collinearorder). contradict. } assert (Out Q W W) by (conclude lemma_ray4). assert (Out Q U P) by (conclude lemma_ray5). assert (CongA A B C P Q W) by (conclude lemma_equalangleshelper). assert (LtA A B C P Q R) by (conclude_def LtA ). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.proposition_03 GeoCoq.Elements.OriginalProofs.lemma_NCorder GeoCoq.Elements.OriginalProofs.lemma_NCdistinct GeoCoq.Elements.OriginalProofs.lemma_ABCequalsCBA GeoCoq.Elements.OriginalProofs.lemma_equalanglestransitive