Library GeoCoq.Elements.OriginalProofs.proposition_28D

Require Export GeoCoq.Elements.OriginalProofs.lemma_NCdistinct.
Require Export GeoCoq.Elements.OriginalProofs.proposition_28A.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinearparallel.
Require Export GeoCoq.Elements.OriginalProofs.lemma_parallelsymmetric.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_28D :
   ∀ B D E G H,
   BetS E G H → CongA E G B G H D → OS B D G H →
   Par G B H D.
Proof.
intros.
assert (nCol G H B) by (conclude_def OS ).
assert (nCol G H D) by (conclude_def OS ).
assert (neq H D) by (forward_using lemma_NCdistinct).
assert (neq D H) by (conclude lemma_inequalitysymmetric).
assert (neq G B) by (forward_using lemma_NCdistinct).
assert (neq B G) by (conclude lemma_inequalitysymmetric).
let Tf:=fresh in
assert (Tf:∃ A, (BetS B G A ∧ Cong G A G B)) by (conclude postulate_extension);destruct Tf as [A];spliter.
assert (BetS A G B) by (conclude axiom_betweennesssymmetry).
let Tf:=fresh in
assert (Tf:∃ C, (BetS D H C ∧ Cong H C H D)) by (conclude postulate_extension);destruct Tf as [C];spliter.
assert (BetS C H D) by (conclude axiom_betweennesssymmetry).
assert (neq G H) by (forward_using lemma_betweennotequal).
assert (neq H G) by (conclude lemma_inequalitysymmetric).
let Tf:=fresh in
assert (Tf:∃ F, (BetS G H F ∧ Cong H F G H)) by (conclude postulate_extension);destruct Tf as [F];spliter.
assert (BetS F H G) by (conclude axiom_betweennesssymmetry).
assert (Par A B C D) by (conclude proposition_28A).
assert (Col D H C) by (conclude_def Col ).
assert (Col C D H) by (forward_using lemma_collinearorder).
assert (neq H D) by (forward_using lemma_NCdistinct).
assert (Par A B H D) by (conclude lemma_collinearparallel).
assert (Par H D A B) by (conclude lemma_parallelsymmetric).
assert (Col B G A) by (conclude_def Col ).
assert (Col A B G) by (forward_using lemma_collinearorder).
assert (Par H D G B) by (conclude lemma_collinearparallel).
assert (Par G B H D) by (conclude lemma_parallelsymmetric).
close.
Qed.

End Euclid.