Library GeoCoq.Elements.OriginalProofs.lemma_TTflip2

Require Export GeoCoq.Elements.OriginalProofs.lemma_betweennotequal.
Require Export GeoCoq.Elements.OriginalProofs.lemma_nullsegment3.
Require Export GeoCoq.Elements.OriginalProofs.lemma_inequalitysymmetric.
Require Export GeoCoq.Elements.OriginalProofs.lemma_congruenceflip.
Require Export GeoCoq.Elements.OriginalProofs.lemma_sumofparts.
Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence2.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma lemma_TTflip2 :
   ∀ A B C D E F G H,
   TT A B C D E F G H →
   TT A B C D H G F E.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ J, (BetS E F J ∧ Cong F J G H ∧ TG A B C D E J)) by (conclude_def TT );destruct Tf as [J];spliter.
let Tf:=fresh in
assert (Tf:∃ K, (BetS A B K ∧ Cong B K C D ∧ Lt E J A K)) by (conclude_def TG );destruct Tf as [K];spliter.
assert (neq F J) by (forward_using lemma_betweennotequal).
assert (neq G H) by (conclude lemma_nullsegment3).
assert (neq H G) by (conclude lemma_inequalitysymmetric).
assert (neq E F) by (forward_using lemma_betweennotequal).
assert (neq F E) by (conclude lemma_inequalitysymmetric).
let Tf:=fresh in
assert (Tf:∃ L, (BetS H G L ∧ Cong G L F E)) by (conclude postulate_extension);destruct Tf as [L];spliter.
assert (Cong L G E F) by (forward_using lemma_congruenceflip).
assert (Cong G H F J) by (conclude lemma_congruencesymmetric).
assert (BetS L G H) by (conclude axiom_betweennesssymmetry).
assert (Cong L H E J) by (conclude lemma_sumofparts).
assert (Cong H L L H) by (conclude cn_equalityreverse).
assert (Cong H L E J) by (conclude lemma_congruencetransitive).
assert (Cong E J H L) by (conclude lemma_congruencesymmetric).
assert (Lt H L A K) by (conclude lemma_lessthancongruence2).
assert (TG A B C D H L) by (conclude_def TG ).
assert (TT A B C D H G F E) by (conclude_def TT ).
close.
Qed.

End Euclid.