Library GeoCoq.Elements.OriginalProofs.proposition_33

Require Export GeoCoq.Elements.OriginalProofs.proposition_29B.
Require Export GeoCoq.Elements.OriginalProofs.proposition_27B.

Section Euclid.

Context `{Ax:euclidean_euclidean}.

Lemma proposition_33 :
   ∀ A B C D M,
   Par A B C D → Cong A B C D → BetS A M D → BetS B M C →
   Par A C B D ∧ Cong A C B D.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ a b c d m, (neq A B ∧ neq C D ∧ Col A B a ∧ Col A B b ∧ neq a b ∧ Col C D c ∧ Col C D d ∧ neq c d ∧ ¬ Meet A B C D ∧ BetS a m d ∧ BetS c m b)) by (conclude_def Par );destruct Tf as [a[b[c[d[m]]]]];spliter.
assert (Col B M C) by (conclude_def Col ).
assert (Col B C M) by (forward_using lemma_collinearorder).
assert (¬ Col B C A).
 {
 intro.
 assert (Col A B C) by (forward_using lemma_collinearorder).
 assert (eq C C) by (conclude cn_equalityreflexive).
 assert (Col C D C) by (conclude_def Col ).
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (TS A B C D) by (conclude_def TS ).
assert (CongA A B C B C D) by (conclude proposition_29B).
assert (¬ Col B C D).
 {
 intro.
 assert (Col C D B) by (forward_using lemma_collinearorder).
 assert (eq B B) by (conclude cn_equalityreflexive).
 assert (Col A B B) by (conclude_def Col ).
 assert (Meet A B C D) by (conclude_def Meet ).
 contradict.
 }
assert (CongA B C D D C B) by (conclude lemma_ABCequalsCBA).
assert (CongA A B C D C B) by (conclude lemma_equalanglestransitive).
assert (Cong B C B C) by (conclude cn_congruencereflexive).
assert (nCol A B C) by (assert (nCol B C A) by auto;
 (forward_using lemma_NCorder)).
assert (Triangle A B C) by (conclude_def Triangle ).
assert (nCol D C B) by (assert (nCol B C D) by auto;forward_using lemma_NCorder).
assert (Triangle D C B) by (conclude_def Triangle ).
assert (Cong B A C D) by (forward_using lemma_congruenceflip).
assert (Cong B C C B) by (forward_using lemma_congruenceflip).
assert ((Cong A C D B ∧ CongA B A C C D B ∧ CongA B C A C B D)) by (conclude proposition_04).
assert (nCol A C B) by (forward_using lemma_NCorder).
assert (CongA A C B B C A) by (conclude lemma_ABCequalsCBA).
assert (CongA A C B C B D) by (conclude lemma_equalanglestransitive).
assert (Cong A C B D) by (forward_using lemma_congruenceflip).
assert (Col C B M) by (forward_using lemma_collinearorder).
assert (nCol C B A) by (forward_using lemma_NCorder).
assert (TS A C B D) by (conclude_def TS ).
assert (Par A C B D) by (conclude proposition_27B).
close.
Qed.

End Euclid.