proposition: Lemma lemma_TTtransitive : forall A B C D E F G H P Q R S, TT A B C D E F G H -> TT E F G H P Q R S -> TT A B C D P Q R S.
proof: Proof. intros. let Tf:=fresh in assert (Tf:exists K, (BetS E F K /\ Cong F K G H /\ TG A B C D E K)) by (conclude_def TT );destruct Tf as [K];spliter. let Tf:=fresh in assert (Tf:exists J, (BetS A B J /\ Cong B J C D /\ Lt E K A J)) by (conclude_def TG );destruct Tf as [J];spliter. let Tf:=fresh in assert (Tf:exists L, (BetS P Q L /\ Cong Q L R S /\ TG E F G H P L)) by (conclude_def TT );destruct Tf as [L];spliter. let Tf:=fresh in assert (Tf:exists M, (BetS E F M /\ Cong F M G H /\ Lt P L E M)) by (conclude_def TG );destruct Tf as [M];spliter. assert (eq K K) by (conclude cn_equalityreflexive). assert (neq F K) by (forward_using lemma_betweennotequal). assert (neq F M) by (forward_using lemma_betweennotequal). assert (Out F K M) by (conclude_def Out ). assert (Out F K K) by (conclude lemma_ray4). assert (Cong G H F M) by (conclude lemma_congruencesymmetric). assert (Cong F K F M) by (conclude lemma_congruencetransitive). assert (eq K M) by (conclude lemma_layoffunique). assert (Lt P L E K) by (conclude cn_equalitysub). assert (Lt P L A J) by (conclude lemma_lessthantransitive). assert (TG A B C D P L) by (conclude_def TG ). assert (TT A B C D P Q R S) by (conclude_def TT ). close. Qed.
imports: GeoCoq.Elements.OriginalProofs.lemma_lessthantransitive