Library GeoCoq.Elements.OriginalProofs.lemma_planeseparation
Require Export GeoCoq.Elements.OriginalProofs.lemma_twolines2.
Require Export GeoCoq.Elements.OriginalProofs.lemma_9_5b.
Require Export GeoCoq.Elements.OriginalProofs.lemma_9_5a.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinear5.
Require Export GeoCoq.Elements.OriginalProofs.proposition_10.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_planeseparation :
∀ A B C D E,
OS C D A B → TS D A B E →
TS C A B E.
Proof.
intros.
rename_H H;
let Tf:=fresh in
assert (Tf:∃ G H Q, (Col A B G ∧ Col A B H ∧ BetS C G Q ∧ BetS D H Q ∧ nCol A B C ∧ nCol A B D)) by (conclude_def OS );destruct Tf as [G[H[Q]]];spliter.
assert (¬ eq A B).
{
intro.
assert (Col A B C) by (conclude_def Col ).
contradict.
}
assert (neq B A) by (conclude lemma_inequalitysymmetric).
let Tf:=fresh in
assert (Tf:∃ W, (BetS D W E ∧ Col A B W ∧ nCol A B D)) by (conclude_def TS );destruct Tf as [W];spliter.
assert (neq C G) by (forward_using lemma_betweennotequal).
assert (neq G C) by (conclude lemma_inequalitysymmetric).
assert (neq G Q) by (forward_using lemma_betweennotequal).
assert (TS C A B E).
by cases on (Col C Q D ∨ nCol C Q D).
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col Q C D) by (forward_using lemma_collinearorder).
assert (¬ neq G H).
{
intro.
assert (neq D Q) by (forward_using lemma_betweennotequal).
assert (Col C G Q) by (conclude_def Col ).
assert (Col D H Q) by (conclude_def Col ).
assert (Col Q C G) by (forward_using lemma_collinearorder).
assert (neq C Q) by (forward_using lemma_betweennotequal).
assert (neq Q C) by (conclude lemma_inequalitysymmetric).
assert (Col C G D) by (conclude lemma_collinear4).
assert (Col D Q H) by (forward_using lemma_collinearorder).
assert (Col D Q C) by (forward_using lemma_collinearorder).
assert (Col Q H C) by (conclude lemma_collinear4).
assert (Col Q C H) by (forward_using lemma_collinearorder).
assert (¬ Col C A B).
{
intro.
assert (Col A B C) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ (Col C A B ∧ Col D A B)).
{
intro.
contradict.
}
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (Col H A B) by (forward_using lemma_collinearorder).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert (Col Q D H) by (forward_using lemma_collinearorder).
assert (Col Q D C) by (forward_using lemma_collinearorder).
assert (neq D Q) by (forward_using lemma_betweennotequal).
assert (neq Q D) by (conclude lemma_inequalitysymmetric).
assert (Col D C H) by (conclude lemma_collinear4).
assert (Col H C D) by (forward_using lemma_collinearorder).
assert (¬ eq C D).
{
intro.
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (eq G H) by (conclude lemma_twolines2).
contradict.
}
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (BetS Q H D) by (conclude axiom_betweennesssymmetry).
assert (BetS Q G D) by (conclude cn_equalitysub).
assert (¬ nCol E D G).
{
intro.
assert (¬ nCol E C G).
{
intro.
assert (¬ BetS G C D).
{
intro.
assert (TS C A B E) by (conclude lemma_9_5b).
contradict.
}
assert (¬ BetS G D C).
{
intro.
assert (¬ Col G C E).
{
intro.
assert (Col E C G) by (forward_using lemma_collinearorder).
contradict.
}
assert (TS C A B E) by (conclude lemma_9_5a).
contradict.
}
assert (eq C D) by (conclude lemma_outerconnectivity).
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (Col C G E) by (forward_using lemma_collinearorder).
assert (Col Q G D) by (conclude_def Col ).
assert (Col Q G C) by (conclude_def Col ).
assert (neq Q G) by (forward_using lemma_betweennotequal).
assert (Col G D C) by (conclude lemma_collinear4).
assert (Col C G D) by (forward_using lemma_collinearorder).
assert (Col C G E) by (forward_using lemma_collinearorder).
assert (neq G C) by (forward_using lemma_betweennotequal).
assert (neq C G) by (conclude lemma_inequalitysymmetric).
assert (Col G D E) by (conclude lemma_collinear4).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col D G E) by (forward_using lemma_collinearorder).
assert (Col D H Q) by (conclude_def Col ).
assert (Col D G Q) by (conclude cn_equalitysub).
assert (neq G D) by (forward_using lemma_betweennotequal).
assert (neq D G) by (conclude lemma_inequalitysymmetric).
assert (Col G E Q) by (conclude lemma_collinear4).
assert (Col G E D) by (forward_using lemma_collinearorder).
assert (Col D W E) by (conclude_def Col ).
assert (Col D E W) by (forward_using lemma_collinearorder).
assert (Col D E G) by (forward_using lemma_collinearorder).
assert (neq D E) by (forward_using lemma_betweennotequal).
assert (Col E W G) by (conclude lemma_collinear4).
assert (Col E W D) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (neq E W) by (conclude lemma_inequalitysymmetric).
assert (Col W G D) by (conclude lemma_collinear4).
assert (Col B W G) by (conclude lemma_collinear4).
assert (Col W G B) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col A W G) by (conclude lemma_collinear4).
assert (Col W G A) by (forward_using lemma_collinearorder).
assert (¬ neq W G).
{
intro.
assert (Col G D B) by (conclude lemma_collinear4).
assert (Col G B D) by (forward_using lemma_collinearorder).
assert (Col G B A) by (forward_using lemma_collinearorder).
assert (¬ neq G B).
{
intro.
assert (Col B D A) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col G D A) by (conclude lemma_collinear4).
assert (Col G A D) by (forward_using lemma_collinearorder).
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (¬ neq G A).
{
intro.
assert (Col A D B) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (eq A G) by (conclude lemma_equalitysymmetric).
assert (eq B G) by (conclude lemma_equalitysymmetric).
assert (eq A B) by (conclude cn_equalitytransitive).
contradict.
}
assert (BetS D G E) by (conclude cn_equalitysub).
assert (BetS E G D) by (conclude axiom_betweennesssymmetry).
assert (¬ BetS G D C).
{
intro.
assert (BetS E G C) by (conclude lemma_3_7b).
assert (BetS C G E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (¬ BetS G C D).
{
intro.
assert (BetS E G C) by (conclude axiom_innertransitivity).
assert (BetS C G E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (¬ BetS C G D).
{
intro.
assert (BetS D G Q) by (conclude cn_equalitysub).
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (BetS D G C) by (conclude axiom_betweennesssymmetry).
assert (neq G D) by (forward_using lemma_betweennotequal).
assert (¬ BetS G D C).
{
intro.
assert (BetS C D G) by (conclude axiom_betweennesssymmetry).
assert (BetS C G C) by (conclude lemma_3_7a).
assert (neq C C) by (forward_using lemma_betweennotequal).
assert (eq C C) by (conclude cn_equalityreflexive).
contradict.
}
assert (¬ BetS G C D).
{
intro.
assert (BetS D G C) by (conclude axiom_betweennesssymmetry).
assert (BetS C G C) by (conclude axiom_innertransitivity).
assert (neq C C) by (forward_using lemma_betweennotequal).
assert (eq C C) by (conclude cn_equalityreflexive).
contradict.
}
assert (BetS Q G D) by (conclude axiom_betweennesssymmetry).
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (eq C D) by (conclude lemma_outerconnectivity).
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (Col Q C D) by (forward_using lemma_collinearorder).
assert (Col Q G C) by (conclude_def Col ).
assert (Col Q C G) by (forward_using lemma_collinearorder).
assert (neq Q C) by (forward_using lemma_betweennotequal).
assert (Col C D G) by (conclude lemma_collinear4).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert ((eq G C ∨ eq G D ∨ eq C D ∨ BetS C G D ∨ BetS G C D ∨ BetS G D C)) by (conclude_def Col ).
assert (TS C A B E).
by cases on (eq G C ∨ eq G D ∨ eq C D ∨ BetS C G D ∨ BetS G C D ∨ BetS G D C).
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col A B C) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col A B D) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (¬ Col Q C D).
{
intro.
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
let Tf:=fresh in
assert (Tf:∃ F, (BetS C F H ∧ BetS D F G)) by (conclude postulate_Pasch_inner);destruct Tf as [F];spliter.
assert (BetS H F C) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
assert (¬ Col E D G).
{
intro.
assert (¬ neq W G).
{
intro.
assert (Col D E G) by (forward_using lemma_collinearorder).
assert (Col D W E) by (conclude_def Col ).
assert (Col B G W) by (conclude lemma_collinear4).
assert (Col W B G) by (forward_using lemma_collinearorder).
assert (Col W B A) by (forward_using lemma_collinearorder).
assert (Col B G A).
by cases on (eq W B ∨ neq W B).
{
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col A G W) by (conclude lemma_collinear4).
assert (Col A G B) by (conclude cn_equalitysub).
assert (Col B G A) by (forward_using lemma_collinearorder).
close.
}
{
assert (Col B G A) by (conclude lemma_collinear4).
close.
}
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (Col E D G) by (forward_using lemma_collinearorder).
assert (Col E D W) by (forward_using lemma_collinearorder).
assert (neq D E) by (forward_using lemma_betweennotequal).
assert (neq E D) by (conclude lemma_inequalitysymmetric).
assert (Col D G W) by (conclude lemma_collinear4).
assert (Col G W D) by (forward_using lemma_collinearorder).
assert (eq B B) by (conclude cn_equalityreflexive).
assert (Col A B B) by (conclude_def Col ).
assert (neq G W) by (conclude lemma_inequalitysymmetric).
assert (Col G W B) by (conclude lemma_collinear5).
assert (Col W D B) by (conclude lemma_collinear4).
assert (Col W B D) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (eq A A) by (conclude cn_equalityreflexive).
assert (Col B A A) by (conclude_def Col ).
assert (Col G W A) by (conclude lemma_collinear5).
assert (Col W D A) by (conclude lemma_collinear4).
assert (neq D W) by (forward_using lemma_betweennotequal).
assert (neq W D) by (conclude lemma_inequalitysymmetric).
assert (Col D B A) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS D G E) by (conclude cn_equalitysub).
assert (BetS E G D) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
assert (BetS E G F) by (conclude axiom_innertransitivity).
assert (¬ Col H C E).
{
intro.
assert (Col H E C) by (forward_using lemma_collinearorder).
assert (Col E H C) by (forward_using lemma_collinearorder).
assert (Col C F H) by (conclude_def Col ).
assert (Col H C F) by (forward_using lemma_collinearorder).
assert (Col H C E) by (forward_using lemma_collinearorder).
assert (neq C H) by (forward_using lemma_betweennotequal).
assert (neq H C) by (conclude lemma_inequalitysymmetric).
assert (Col C F E) by (conclude lemma_collinear4).
assert (Col E F C) by (forward_using lemma_collinearorder).
assert (BetS D F E) by (conclude lemma_3_6b).
assert (Col D F E) by (conclude_def Col ).
assert (Col E F D) by (forward_using lemma_collinearorder).
assert (neq F E) by (forward_using lemma_betweennotequal).
assert (neq E F) by (conclude lemma_inequalitysymmetric).
assert (Col F C D) by (conclude lemma_collinear4).
assert (Col F D C) by (forward_using lemma_collinearorder).
assert (Col D F G) by (conclude_def Col ).
assert (Col F D G) by (forward_using lemma_collinearorder).
assert (neq D F) by (forward_using lemma_betweennotequal).
assert (neq F D) by (conclude lemma_inequalitysymmetric).
assert (Col D C G) by (conclude lemma_collinear4).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert (Col C G Q) by (conclude_def Col ).
assert (Col G C Q) by (forward_using lemma_collinearorder).
assert (neq C G) by (forward_using lemma_betweennotequal).
assert (neq G C) by (conclude lemma_inequalitysymmetric).
assert (Col C D Q) by (conclude lemma_collinear4).
assert (Col Q C D) by (forward_using lemma_collinearorder).
contradict.
}
let Tf:=fresh in
assert (Tf:∃ M, (BetS E M C ∧ BetS H G M)) by (conclude postulate_Pasch_outer);destruct Tf as [M];spliter.
assert (Col H G M) by (conclude_def Col ).
assert (Col B H G) by (conclude lemma_collinear4).
assert (Col H G B) by (forward_using lemma_collinearorder).
assert (neq H G) by (forward_using lemma_betweennotequal).
assert (Col G M B) by (conclude lemma_collinear4).
assert (Col G B M) by (forward_using lemma_collinearorder).
assert (Col G B A) by (forward_using lemma_collinearorder).
assert (Col A B M).
by cases on (eq B G ∨ neq B G).
{
assert (Col B A H) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col A H G) by (conclude lemma_collinear4).
assert (Col H G A) by (forward_using lemma_collinearorder).
assert (Col G A M) by (conclude lemma_collinear4).
assert (Col B A M) by (conclude cn_equalitysub).
assert (Col A B M) by (forward_using lemma_collinearorder).
close.
}
{
assert (neq G B) by (conclude lemma_inequalitysymmetric).
assert (Col B M A) by (conclude lemma_collinear4).
assert (Col A B M) by (forward_using lemma_collinearorder).
close.
}
assert (BetS C M E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (TS F A B E) by (conclude lemma_9_5b).
assert (¬ eq G H).
{
intro.
assert (Col D H Q) by (conclude_def Col ).
assert (Col C G Q) by (conclude_def Col ).
assert (Col Q H D) by (forward_using lemma_collinearorder).
assert (Col Q G C) by (forward_using lemma_collinearorder).
assert (Col Q H C) by (conclude cn_equalitysub).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (neq Q H) by (conclude lemma_inequalitysymmetric).
assert (Col H D C) by (conclude lemma_collinear4).
assert (Col H D Q) by (forward_using lemma_collinearorder).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (neq H D) by (conclude lemma_inequalitysymmetric).
assert (Col D C Q) by (conclude lemma_collinear4).
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ Col H C E).
{
intro.
let Tf:=fresh in
assert (Tf:∃ K, (BetS G K H ∧ Cong K G K H)) by (conclude proposition_10);destruct Tf as [K];spliter.
assert (¬ Col G D E).
{
intro.
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS E W D) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
let Tf:=fresh in
assert (Tf:∃ J, (BetS E J F ∧ BetS G J W)) by (conclude postulate_Pasch_inner);destruct Tf as [J];spliter.
assert (Col G J W) by (conclude_def Col ).
assert (Col E F J) by (conclude_def Col ).
assert (Col E C H) by (forward_using lemma_collinearorder).
assert (Col F E J) by (forward_using lemma_collinearorder).
assert (Col C F H) by (conclude_def Col ).
assert (Col C H F) by (forward_using lemma_collinearorder).
assert (Col C H E) by (forward_using lemma_collinearorder).
assert (neq C H) by (forward_using lemma_betweennotequal).
assert (Col H F E) by (conclude lemma_collinear4).
assert (Col F E H) by (forward_using lemma_collinearorder).
assert (neq E F) by (forward_using lemma_betweennotequal).
assert (neq F E) by (conclude lemma_inequalitysymmetric).
assert (Col E J H) by (conclude lemma_collinear4).
assert (Col E H J) by (forward_using lemma_collinearorder).
assert (Col G W J) by (forward_using lemma_collinearorder).
assert (eq H H) by (conclude cn_equalityreflexive).
assert (Col E H H) by (conclude_def Col ).
assert (Col G H W) by (conclude lemma_collinear5).
assert (Col G W H) by (forward_using lemma_collinearorder).
assert (¬ eq G W).
{
intro.
assert (Col D W E) by (conclude_def Col ).
assert (Col D G E) by (conclude cn_equalitysub).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col E J H) by (conclude lemma_collinear4).
assert (Col W J H) by (conclude lemma_collinear4).
assert (Col J H E) by (forward_using lemma_collinearorder).
assert (Col J H W) by (forward_using lemma_collinearorder).
assert (¬ eq H W).
{
intro.
assert (Col D W E) by (conclude_def Col ).
assert (Col D H E) by (conclude cn_equalitysub).
assert (Col D H Q) by (conclude_def Col ).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (Col H E Q) by (conclude lemma_collinear4).
assert (Col H E C) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (neq H E) by (conclude cn_equalitysub).
assert (Col E Q C) by (conclude lemma_collinear4).
assert (Col E C Q) by (forward_using lemma_collinearorder).
assert (Col E C H) by (forward_using lemma_collinearorder).
assert (¬ neq E C).
{
intro.
assert (Col C Q H) by (conclude lemma_collinear4).
assert (Col H Q C) by (forward_using lemma_collinearorder).
assert (Col H Q D) by (forward_using lemma_collinearorder).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (Col Q C D) by (conclude lemma_collinear4).
contradict.
}
assert (Col E W D) by (forward_using lemma_collinearorder).
assert (Col C W D) by (conclude cn_equalitysub).
assert (Col C H D) by (conclude cn_equalitysub).
assert (Col D H C) by (forward_using lemma_collinearorder).
assert (Col D H Q) by (conclude_def Col ).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (Col H C Q) by (conclude lemma_collinear4).
assert (Col H Q C) by (forward_using lemma_collinearorder).
assert (Col H Q D) by (forward_using lemma_collinearorder).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (Col Q C D) by (conclude lemma_collinear4).
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ neq J H).
{
intro.
assert (Col H E W) by (conclude lemma_collinear4).
assert (Col H W E) by (forward_using lemma_collinearorder).
assert (Col H W G) by (forward_using lemma_collinearorder).
assert (Col W E G) by (conclude lemma_collinear4).
assert (Col D W E) by (conclude_def Col ).
assert (Col W E D) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (Col E G D) by (conclude lemma_collinear4).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS E H F) by (conclude cn_equalitysub).
assert (BetS F H E) by (conclude axiom_betweennesssymmetry).
assert (BetS C H E) by (conclude lemma_3_7a).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (TS C A B E) by (conclude lemma_9_5a).
contradict.
}
close.
}
close.
Qed.
End Euclid.
Require Export GeoCoq.Elements.OriginalProofs.lemma_9_5b.
Require Export GeoCoq.Elements.OriginalProofs.lemma_9_5a.
Require Export GeoCoq.Elements.OriginalProofs.lemma_collinear5.
Require Export GeoCoq.Elements.OriginalProofs.proposition_10.
Section Euclid.
Context `{Ax:euclidean_neutral_ruler_compass}.
Lemma lemma_planeseparation :
∀ A B C D E,
OS C D A B → TS D A B E →
TS C A B E.
Proof.
intros.
rename_H H;
let Tf:=fresh in
assert (Tf:∃ G H Q, (Col A B G ∧ Col A B H ∧ BetS C G Q ∧ BetS D H Q ∧ nCol A B C ∧ nCol A B D)) by (conclude_def OS );destruct Tf as [G[H[Q]]];spliter.
assert (¬ eq A B).
{
intro.
assert (Col A B C) by (conclude_def Col ).
contradict.
}
assert (neq B A) by (conclude lemma_inequalitysymmetric).
let Tf:=fresh in
assert (Tf:∃ W, (BetS D W E ∧ Col A B W ∧ nCol A B D)) by (conclude_def TS );destruct Tf as [W];spliter.
assert (neq C G) by (forward_using lemma_betweennotequal).
assert (neq G C) by (conclude lemma_inequalitysymmetric).
assert (neq G Q) by (forward_using lemma_betweennotequal).
assert (TS C A B E).
by cases on (Col C Q D ∨ nCol C Q D).
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col Q C D) by (forward_using lemma_collinearorder).
assert (¬ neq G H).
{
intro.
assert (neq D Q) by (forward_using lemma_betweennotequal).
assert (Col C G Q) by (conclude_def Col ).
assert (Col D H Q) by (conclude_def Col ).
assert (Col Q C G) by (forward_using lemma_collinearorder).
assert (neq C Q) by (forward_using lemma_betweennotequal).
assert (neq Q C) by (conclude lemma_inequalitysymmetric).
assert (Col C G D) by (conclude lemma_collinear4).
assert (Col D Q H) by (forward_using lemma_collinearorder).
assert (Col D Q C) by (forward_using lemma_collinearorder).
assert (Col Q H C) by (conclude lemma_collinear4).
assert (Col Q C H) by (forward_using lemma_collinearorder).
assert (¬ Col C A B).
{
intro.
assert (Col A B C) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ (Col C A B ∧ Col D A B)).
{
intro.
contradict.
}
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (Col H A B) by (forward_using lemma_collinearorder).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert (Col Q D H) by (forward_using lemma_collinearorder).
assert (Col Q D C) by (forward_using lemma_collinearorder).
assert (neq D Q) by (forward_using lemma_betweennotequal).
assert (neq Q D) by (conclude lemma_inequalitysymmetric).
assert (Col D C H) by (conclude lemma_collinear4).
assert (Col H C D) by (forward_using lemma_collinearorder).
assert (¬ eq C D).
{
intro.
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (eq G H) by (conclude lemma_twolines2).
contradict.
}
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (BetS Q H D) by (conclude axiom_betweennesssymmetry).
assert (BetS Q G D) by (conclude cn_equalitysub).
assert (¬ nCol E D G).
{
intro.
assert (¬ nCol E C G).
{
intro.
assert (¬ BetS G C D).
{
intro.
assert (TS C A B E) by (conclude lemma_9_5b).
contradict.
}
assert (¬ BetS G D C).
{
intro.
assert (¬ Col G C E).
{
intro.
assert (Col E C G) by (forward_using lemma_collinearorder).
contradict.
}
assert (TS C A B E) by (conclude lemma_9_5a).
contradict.
}
assert (eq C D) by (conclude lemma_outerconnectivity).
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (Col C G E) by (forward_using lemma_collinearorder).
assert (Col Q G D) by (conclude_def Col ).
assert (Col Q G C) by (conclude_def Col ).
assert (neq Q G) by (forward_using lemma_betweennotequal).
assert (Col G D C) by (conclude lemma_collinear4).
assert (Col C G D) by (forward_using lemma_collinearorder).
assert (Col C G E) by (forward_using lemma_collinearorder).
assert (neq G C) by (forward_using lemma_betweennotequal).
assert (neq C G) by (conclude lemma_inequalitysymmetric).
assert (Col G D E) by (conclude lemma_collinear4).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col D G E) by (forward_using lemma_collinearorder).
assert (Col D H Q) by (conclude_def Col ).
assert (Col D G Q) by (conclude cn_equalitysub).
assert (neq G D) by (forward_using lemma_betweennotequal).
assert (neq D G) by (conclude lemma_inequalitysymmetric).
assert (Col G E Q) by (conclude lemma_collinear4).
assert (Col G E D) by (forward_using lemma_collinearorder).
assert (Col D W E) by (conclude_def Col ).
assert (Col D E W) by (forward_using lemma_collinearorder).
assert (Col D E G) by (forward_using lemma_collinearorder).
assert (neq D E) by (forward_using lemma_betweennotequal).
assert (Col E W G) by (conclude lemma_collinear4).
assert (Col E W D) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (neq E W) by (conclude lemma_inequalitysymmetric).
assert (Col W G D) by (conclude lemma_collinear4).
assert (Col B W G) by (conclude lemma_collinear4).
assert (Col W G B) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col A W G) by (conclude lemma_collinear4).
assert (Col W G A) by (forward_using lemma_collinearorder).
assert (¬ neq W G).
{
intro.
assert (Col G D B) by (conclude lemma_collinear4).
assert (Col G B D) by (forward_using lemma_collinearorder).
assert (Col G B A) by (forward_using lemma_collinearorder).
assert (¬ neq G B).
{
intro.
assert (Col B D A) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col G D A) by (conclude lemma_collinear4).
assert (Col G A D) by (forward_using lemma_collinearorder).
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (¬ neq G A).
{
intro.
assert (Col A D B) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (eq A G) by (conclude lemma_equalitysymmetric).
assert (eq B G) by (conclude lemma_equalitysymmetric).
assert (eq A B) by (conclude cn_equalitytransitive).
contradict.
}
assert (BetS D G E) by (conclude cn_equalitysub).
assert (BetS E G D) by (conclude axiom_betweennesssymmetry).
assert (¬ BetS G D C).
{
intro.
assert (BetS E G C) by (conclude lemma_3_7b).
assert (BetS C G E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (¬ BetS G C D).
{
intro.
assert (BetS E G C) by (conclude axiom_innertransitivity).
assert (BetS C G E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (¬ BetS C G D).
{
intro.
assert (BetS D G Q) by (conclude cn_equalitysub).
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (BetS D G C) by (conclude axiom_betweennesssymmetry).
assert (neq G D) by (forward_using lemma_betweennotequal).
assert (¬ BetS G D C).
{
intro.
assert (BetS C D G) by (conclude axiom_betweennesssymmetry).
assert (BetS C G C) by (conclude lemma_3_7a).
assert (neq C C) by (forward_using lemma_betweennotequal).
assert (eq C C) by (conclude cn_equalityreflexive).
contradict.
}
assert (¬ BetS G C D).
{
intro.
assert (BetS D G C) by (conclude axiom_betweennesssymmetry).
assert (BetS C G C) by (conclude axiom_innertransitivity).
assert (neq C C) by (forward_using lemma_betweennotequal).
assert (eq C C) by (conclude cn_equalityreflexive).
contradict.
}
assert (BetS Q G D) by (conclude axiom_betweennesssymmetry).
assert (BetS Q G C) by (conclude axiom_betweennesssymmetry).
assert (eq C D) by (conclude lemma_outerconnectivity).
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
assert (Col Q C D) by (forward_using lemma_collinearorder).
assert (Col Q G C) by (conclude_def Col ).
assert (Col Q C G) by (forward_using lemma_collinearorder).
assert (neq Q C) by (forward_using lemma_betweennotequal).
assert (Col C D G) by (conclude lemma_collinear4).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert ((eq G C ∨ eq G D ∨ eq C D ∨ BetS C G D ∨ BetS G C D ∨ BetS G D C)) by (conclude_def Col ).
assert (TS C A B E).
by cases on (eq G C ∨ eq G D ∨ eq C D ∨ BetS C G D ∨ BetS G C D ∨ BetS G D C).
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col A B C) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (Col A B D) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (TS C A B E) by (conclude cn_equalitysub).
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
contradict.
}
close.
}
contradict.
}
close.
}
{
assert (¬ ¬ TS C A B E).
{
intro.
assert (¬ Col Q C D).
{
intro.
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
let Tf:=fresh in
assert (Tf:∃ F, (BetS C F H ∧ BetS D F G)) by (conclude postulate_Pasch_inner);destruct Tf as [F];spliter.
assert (BetS H F C) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
assert (¬ Col E D G).
{
intro.
assert (¬ neq W G).
{
intro.
assert (Col D E G) by (forward_using lemma_collinearorder).
assert (Col D W E) by (conclude_def Col ).
assert (Col B G W) by (conclude lemma_collinear4).
assert (Col W B G) by (forward_using lemma_collinearorder).
assert (Col W B A) by (forward_using lemma_collinearorder).
assert (Col B G A).
by cases on (eq W B ∨ neq W B).
{
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col A G W) by (conclude lemma_collinear4).
assert (Col A G B) by (conclude cn_equalitysub).
assert (Col B G A) by (forward_using lemma_collinearorder).
close.
}
{
assert (Col B G A) by (conclude lemma_collinear4).
close.
}
assert (Col G A B) by (forward_using lemma_collinearorder).
assert (Col E D G) by (forward_using lemma_collinearorder).
assert (Col E D W) by (forward_using lemma_collinearorder).
assert (neq D E) by (forward_using lemma_betweennotequal).
assert (neq E D) by (conclude lemma_inequalitysymmetric).
assert (Col D G W) by (conclude lemma_collinear4).
assert (Col G W D) by (forward_using lemma_collinearorder).
assert (eq B B) by (conclude cn_equalityreflexive).
assert (Col A B B) by (conclude_def Col ).
assert (neq G W) by (conclude lemma_inequalitysymmetric).
assert (Col G W B) by (conclude lemma_collinear5).
assert (Col W D B) by (conclude lemma_collinear4).
assert (Col W B D) by (forward_using lemma_collinearorder).
assert (Col B A W) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (eq A A) by (conclude cn_equalityreflexive).
assert (Col B A A) by (conclude_def Col ).
assert (Col G W A) by (conclude lemma_collinear5).
assert (Col W D A) by (conclude lemma_collinear4).
assert (neq D W) by (forward_using lemma_betweennotequal).
assert (neq W D) by (conclude lemma_inequalitysymmetric).
assert (Col D B A) by (conclude lemma_collinear4).
assert (Col A B D) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS D G E) by (conclude cn_equalitysub).
assert (BetS E G D) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
assert (BetS E G F) by (conclude axiom_innertransitivity).
assert (¬ Col H C E).
{
intro.
assert (Col H E C) by (forward_using lemma_collinearorder).
assert (Col E H C) by (forward_using lemma_collinearorder).
assert (Col C F H) by (conclude_def Col ).
assert (Col H C F) by (forward_using lemma_collinearorder).
assert (Col H C E) by (forward_using lemma_collinearorder).
assert (neq C H) by (forward_using lemma_betweennotequal).
assert (neq H C) by (conclude lemma_inequalitysymmetric).
assert (Col C F E) by (conclude lemma_collinear4).
assert (Col E F C) by (forward_using lemma_collinearorder).
assert (BetS D F E) by (conclude lemma_3_6b).
assert (Col D F E) by (conclude_def Col ).
assert (Col E F D) by (forward_using lemma_collinearorder).
assert (neq F E) by (forward_using lemma_betweennotequal).
assert (neq E F) by (conclude lemma_inequalitysymmetric).
assert (Col F C D) by (conclude lemma_collinear4).
assert (Col F D C) by (forward_using lemma_collinearorder).
assert (Col D F G) by (conclude_def Col ).
assert (Col F D G) by (forward_using lemma_collinearorder).
assert (neq D F) by (forward_using lemma_betweennotequal).
assert (neq F D) by (conclude lemma_inequalitysymmetric).
assert (Col D C G) by (conclude lemma_collinear4).
assert (Col G C D) by (forward_using lemma_collinearorder).
assert (Col C G Q) by (conclude_def Col ).
assert (Col G C Q) by (forward_using lemma_collinearorder).
assert (neq C G) by (forward_using lemma_betweennotequal).
assert (neq G C) by (conclude lemma_inequalitysymmetric).
assert (Col C D Q) by (conclude lemma_collinear4).
assert (Col Q C D) by (forward_using lemma_collinearorder).
contradict.
}
let Tf:=fresh in
assert (Tf:∃ M, (BetS E M C ∧ BetS H G M)) by (conclude postulate_Pasch_outer);destruct Tf as [M];spliter.
assert (Col H G M) by (conclude_def Col ).
assert (Col B H G) by (conclude lemma_collinear4).
assert (Col H G B) by (forward_using lemma_collinearorder).
assert (neq H G) by (forward_using lemma_betweennotequal).
assert (Col G M B) by (conclude lemma_collinear4).
assert (Col G B M) by (forward_using lemma_collinearorder).
assert (Col G B A) by (forward_using lemma_collinearorder).
assert (Col A B M).
by cases on (eq B G ∨ neq B G).
{
assert (Col B A H) by (forward_using lemma_collinearorder).
assert (Col B A G) by (forward_using lemma_collinearorder).
assert (Col A H G) by (conclude lemma_collinear4).
assert (Col H G A) by (forward_using lemma_collinearorder).
assert (Col G A M) by (conclude lemma_collinear4).
assert (Col B A M) by (conclude cn_equalitysub).
assert (Col A B M) by (forward_using lemma_collinearorder).
close.
}
{
assert (neq G B) by (conclude lemma_inequalitysymmetric).
assert (Col B M A) by (conclude lemma_collinear4).
assert (Col A B M) by (forward_using lemma_collinearorder).
close.
}
assert (BetS C M E) by (conclude axiom_betweennesssymmetry).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (TS F A B E) by (conclude lemma_9_5b).
assert (¬ eq G H).
{
intro.
assert (Col D H Q) by (conclude_def Col ).
assert (Col C G Q) by (conclude_def Col ).
assert (Col Q H D) by (forward_using lemma_collinearorder).
assert (Col Q G C) by (forward_using lemma_collinearorder).
assert (Col Q H C) by (conclude cn_equalitysub).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (neq Q H) by (conclude lemma_inequalitysymmetric).
assert (Col H D C) by (conclude lemma_collinear4).
assert (Col H D Q) by (forward_using lemma_collinearorder).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (neq H D) by (conclude lemma_inequalitysymmetric).
assert (Col D C Q) by (conclude lemma_collinear4).
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ Col H C E).
{
intro.
let Tf:=fresh in
assert (Tf:∃ K, (BetS G K H ∧ Cong K G K H)) by (conclude proposition_10);destruct Tf as [K];spliter.
assert (¬ Col G D E).
{
intro.
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS E W D) by (conclude axiom_betweennesssymmetry).
assert (BetS G F D) by (conclude axiom_betweennesssymmetry).
let Tf:=fresh in
assert (Tf:∃ J, (BetS E J F ∧ BetS G J W)) by (conclude postulate_Pasch_inner);destruct Tf as [J];spliter.
assert (Col G J W) by (conclude_def Col ).
assert (Col E F J) by (conclude_def Col ).
assert (Col E C H) by (forward_using lemma_collinearorder).
assert (Col F E J) by (forward_using lemma_collinearorder).
assert (Col C F H) by (conclude_def Col ).
assert (Col C H F) by (forward_using lemma_collinearorder).
assert (Col C H E) by (forward_using lemma_collinearorder).
assert (neq C H) by (forward_using lemma_betweennotequal).
assert (Col H F E) by (conclude lemma_collinear4).
assert (Col F E H) by (forward_using lemma_collinearorder).
assert (neq E F) by (forward_using lemma_betweennotequal).
assert (neq F E) by (conclude lemma_inequalitysymmetric).
assert (Col E J H) by (conclude lemma_collinear4).
assert (Col E H J) by (forward_using lemma_collinearorder).
assert (Col G W J) by (forward_using lemma_collinearorder).
assert (eq H H) by (conclude cn_equalityreflexive).
assert (Col E H H) by (conclude_def Col ).
assert (Col G H W) by (conclude lemma_collinear5).
assert (Col G W H) by (forward_using lemma_collinearorder).
assert (¬ eq G W).
{
intro.
assert (Col D W E) by (conclude_def Col ).
assert (Col D G E) by (conclude cn_equalitysub).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (Col E J H) by (conclude lemma_collinear4).
assert (Col W J H) by (conclude lemma_collinear4).
assert (Col J H E) by (forward_using lemma_collinearorder).
assert (Col J H W) by (forward_using lemma_collinearorder).
assert (¬ eq H W).
{
intro.
assert (Col D W E) by (conclude_def Col ).
assert (Col D H E) by (conclude cn_equalitysub).
assert (Col D H Q) by (conclude_def Col ).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (Col H E Q) by (conclude lemma_collinear4).
assert (Col H E C) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (neq H E) by (conclude cn_equalitysub).
assert (Col E Q C) by (conclude lemma_collinear4).
assert (Col E C Q) by (forward_using lemma_collinearorder).
assert (Col E C H) by (forward_using lemma_collinearorder).
assert (¬ neq E C).
{
intro.
assert (Col C Q H) by (conclude lemma_collinear4).
assert (Col H Q C) by (forward_using lemma_collinearorder).
assert (Col H Q D) by (forward_using lemma_collinearorder).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (Col Q C D) by (conclude lemma_collinear4).
contradict.
}
assert (Col E W D) by (forward_using lemma_collinearorder).
assert (Col C W D) by (conclude cn_equalitysub).
assert (Col C H D) by (conclude cn_equalitysub).
assert (Col D H C) by (forward_using lemma_collinearorder).
assert (Col D H Q) by (conclude_def Col ).
assert (neq D H) by (forward_using lemma_betweennotequal).
assert (Col H C Q) by (conclude lemma_collinear4).
assert (Col H Q C) by (forward_using lemma_collinearorder).
assert (Col H Q D) by (forward_using lemma_collinearorder).
assert (neq H Q) by (forward_using lemma_betweennotequal).
assert (Col Q C D) by (conclude lemma_collinear4).
assert (Col C Q D) by (forward_using lemma_collinearorder).
contradict.
}
assert (¬ neq J H).
{
intro.
assert (Col H E W) by (conclude lemma_collinear4).
assert (Col H W E) by (forward_using lemma_collinearorder).
assert (Col H W G) by (forward_using lemma_collinearorder).
assert (Col W E G) by (conclude lemma_collinear4).
assert (Col D W E) by (conclude_def Col ).
assert (Col W E D) by (forward_using lemma_collinearorder).
assert (neq W E) by (forward_using lemma_betweennotequal).
assert (Col E G D) by (conclude lemma_collinear4).
assert (Col E D G) by (forward_using lemma_collinearorder).
contradict.
}
assert (BetS E H F) by (conclude cn_equalitysub).
assert (BetS F H E) by (conclude axiom_betweennesssymmetry).
assert (BetS C H E) by (conclude lemma_3_7a).
assert (TS C A B E) by (conclude_def TS ).
contradict.
}
assert (TS C A B E) by (conclude lemma_9_5a).
contradict.
}
close.
}
close.
Qed.
End Euclid.