Theorem btwnconn1lem12 36062

Description: Lemma for btwnconn1 36065. Using a long string of invocations of linecgr 36045, we show that 𝐷 = 𝑑. (Contributed by Scott Fenton, 9-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem12

⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐷 = 𝑑)

Proof of Theorem btwnconn1lem12

Step	Hyp	Ref	Expression
1		simp11 1203	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
2		simp2l1 1272	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
3		simp2l3 1274	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑐 ∈ (𝔼‘𝑁))
4		simp31 1209	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
5		simp32 1210	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑄 ∈ (𝔼‘𝑁))
6		simp1l3 1268	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐶 ≠ 𝑐)
7	6	ad2antrl 727	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 ≠ 𝑐)
8		simpr1l 1230	. . . . . . 7 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → 𝐶 Btwn ⟨𝑐, 𝑃⟩)
9	8	ad2antll 728	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑐, 𝑃⟩)
10		btwncolinear5 36037	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑐 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝑐, 𝑃⟩ → 𝐶 Colinear ⟨𝑐, 𝑃⟩))
11	1, 3, 4, 2, 10	syl13anc 1372	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝑐, 𝑃⟩ → 𝐶 Colinear ⟨𝑐, 𝑃⟩))
12	11	adantr 480	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐶 Btwn ⟨𝑐, 𝑃⟩ → 𝐶 Colinear ⟨𝑐, 𝑃⟩))
13	9, 12	mpd 15	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Colinear ⟨𝑐, 𝑃⟩)
14		simp2l2 1273	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐷 ∈ (𝔼‘𝑁))
15		simp2r1 1275	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑑 ∈ (𝔼‘𝑁))
16		simpr1r 1231	. . . . . . . 8 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)
17	16	ad2antll 728	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩)
18		simp2rr 1243	. . . . . . . 8 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)
19	18	ad2antrl 727	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)
20	1, 2, 4, 2, 15, 2, 14, 17, 19	cgrtrand 35957	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝐷⟩)
21		btwnconn1lem11 36061	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐷, 𝐶⟩Cgr⟨𝑄, 𝐶⟩)
22	1, 14, 2, 5, 2, 21	cgrcomlrand 35965	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝐷⟩Cgr⟨𝐶, 𝑄⟩)
23	1, 2, 4, 2, 14, 2, 5, 20, 22	cgrtrand 35957	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑄⟩)
24		simp12 1204	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
25		simp13 1205	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
26		simp2r2 1276	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑏 ∈ (𝔼‘𝑁))
27		simp1rr 1239	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐵 Btwn ⟨𝐴, 𝐷⟩)
28	27	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐵 Btwn ⟨𝐴, 𝐷⟩)
29		simp2ll 1240	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐷 Btwn ⟨𝐴, 𝑐⟩)
30	29	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐷 Btwn ⟨𝐴, 𝑐⟩)
31	1, 24, 25, 14, 3, 28, 30	btwnexchand 35990	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐵 Btwn ⟨𝐴, 𝑐⟩)
32		simp3ll 1244	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝑐 Btwn ⟨𝐴, 𝑏⟩)
33	32	ad2antrl 727	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑐 Btwn ⟨𝐴, 𝑏⟩)
34	1, 24, 25, 3, 26, 31, 33	btwnexch3and 35985	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑐 Btwn ⟨𝐵, 𝑏⟩)
35		opeq1 4897	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝑏 → ⟨𝐵, 𝑏⟩ = ⟨𝑏, 𝑏⟩)
36	35	breq2d 5178	. . . . . . . . . . . . 13 ⊢ (𝐵 = 𝑏 → (𝑐 Btwn ⟨𝐵, 𝑏⟩ ↔ 𝑐 Btwn ⟨𝑏, 𝑏⟩))
37	36	biimpac 478	. . . . . . . . . . . 12 ⊢ ((𝑐 Btwn ⟨𝐵, 𝑏⟩ ∧ 𝐵 = 𝑏) → 𝑐 Btwn ⟨𝑏, 𝑏⟩)
38		axbtwnid 28972	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑐 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (𝑐 Btwn ⟨𝑏, 𝑏⟩ → 𝑐 = 𝑏))
39	1, 3, 26, 38	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑐 Btwn ⟨𝑏, 𝑏⟩ → 𝑐 = 𝑏))
40	37, 39	syl5 34	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝑐 Btwn ⟨𝐵, 𝑏⟩ ∧ 𝐵 = 𝑏) → 𝑐 = 𝑏))
41	40	expd 415	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑐 Btwn ⟨𝐵, 𝑏⟩ → (𝐵 = 𝑏 → 𝑐 = 𝑏)))
42	41	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝑐 Btwn ⟨𝐵, 𝑏⟩ → (𝐵 = 𝑏 → 𝑐 = 𝑏)))
43	34, 42	mpd 15	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐵 = 𝑏 → 𝑐 = 𝑏))
44		simp1 1136	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
45		simp2l 1199	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)))
46		simp2r 1200	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)))
47	44, 45, 46	3jca 1128	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))))
48		simpl 482	. . . . . . . . . . 11 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))))
49		simprl 770	. . . . . . . . . . 11 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))
50	48, 49	jca 511	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))) → ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩)))
51		btwnconn1lem7 36057	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → 𝐶 ≠ 𝑑)
52	47, 50, 51	syl2an 595	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 ≠ 𝑑)
53		simp2rl 1242	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐶 Btwn ⟨𝐴, 𝑑⟩)
54	53	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝐴, 𝑑⟩)
55		simp3rl 1246	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝑑 Btwn ⟨𝐴, 𝑏⟩)
56	55	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑑 Btwn ⟨𝐴, 𝑏⟩)
57	1, 24, 2, 15, 26, 54, 56	btwnexch3and 35985	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑑 Btwn ⟨𝐶, 𝑏⟩)
58		btwncolinear2 36034	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁))) → (𝑑 Btwn ⟨𝐶, 𝑏⟩ → 𝐶 Colinear ⟨𝑑, 𝑏⟩))
59	1, 2, 26, 15, 58	syl13anc 1372	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑑 Btwn ⟨𝐶, 𝑏⟩ → 𝐶 Colinear ⟨𝑑, 𝑏⟩))
60	59	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝑑 Btwn ⟨𝐶, 𝑏⟩ → 𝐶 Colinear ⟨𝑑, 𝑏⟩))
61	57, 60	mpd 15	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Colinear ⟨𝑑, 𝑏⟩)
62		simp33 1211	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝑅 ∈ (𝔼‘𝑁))
63		simpr2r 1233	. . . . . . . . . . . . . 14 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)
64	63	ad2antll 728	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩)
65		btwnconn1lem5 36055	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ (𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩))) → ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩)
66	47, 50, 65	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩)
67		opeq2 4898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 = 𝐶 → ⟨𝐶, 𝑅⟩ = ⟨𝐶, 𝐶⟩)
68	67	breq1d 5176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 = 𝐶 → (⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ↔ ⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩))
69	68	anbi1d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 = 𝐶 → ((⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) ↔ (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩)))
70	69	biimpac 478	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) ∧ 𝑅 = 𝐶) → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩))
71		simp2r3 1277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → 𝐸 ∈ (𝔼‘𝑁))
72		cgrid2 35967	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩ → 𝐶 = 𝐸))
73	1, 2, 2, 71, 72	syl13anc 1372	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩ → 𝐶 = 𝐸))
74		opeq1 4897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 = 𝐸 → ⟨𝐶, 𝐶⟩ = ⟨𝐸, 𝐶⟩)
75		opeq1 4897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 = 𝐸 → ⟨𝐶, 𝑐⟩ = ⟨𝐸, 𝑐⟩)
76	74, 75	breq12d 5179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 = 𝐸 → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝑐⟩ ↔ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩))
77	76	biimpar 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 = 𝐸 ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) → ⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝑐⟩)
78		cgrid2 35967	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝑐⟩ → 𝐶 = 𝑐))
79	1, 2, 2, 3, 78	syl13anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝑐⟩ → 𝐶 = 𝑐))
80	77, 79	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝐶 = 𝐸 ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) → 𝐶 = 𝑐))
81	73, 80	syland 602	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((⟨𝐶, 𝐶⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) → 𝐶 = 𝑐))
82	70, 81	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (((⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) ∧ 𝑅 = 𝐶) → 𝐶 = 𝑐))
83	82	expd 415	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) → (𝑅 = 𝐶 → 𝐶 = 𝑐)))
84	83	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ((⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩ ∧ ⟨𝐸, 𝐶⟩Cgr⟨𝐸, 𝑐⟩) → (𝑅 = 𝐶 → 𝐶 = 𝑐)))
85	64, 66, 84	mp2and 698	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝑅 = 𝐶 → 𝐶 = 𝑐))
86	85	necon3d 2967	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐶 ≠ 𝑐 → 𝑅 ≠ 𝐶))
87	7, 86	mpd 15	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑅 ≠ 𝐶)
88		simpr2l 1232	. . . . . . . . . . . 12 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → 𝐶 Btwn ⟨𝑑, 𝑅⟩)
89	88	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝑑, 𝑅⟩)
90		btwncolinear4 36036	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝑑, 𝑅⟩ → 𝑅 Colinear ⟨𝐶, 𝑑⟩))
91	1, 15, 62, 2, 90	syl13anc 1372	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝑑, 𝑅⟩ → 𝑅 Colinear ⟨𝐶, 𝑑⟩))
92	91	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐶 Btwn ⟨𝑑, 𝑅⟩ → 𝑅 Colinear ⟨𝐶, 𝑑⟩))
93	89, 92	mpd 15	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑅 Colinear ⟨𝐶, 𝑑⟩)
94		simpr3r 1235	. . . . . . . . . . . 12 ⊢ (((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))) → ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)
95	94	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)
96	1, 62, 5, 62, 4, 95	cgrcomand 35955	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑅, 𝑃⟩Cgr⟨𝑅, 𝑄⟩)
97	1, 62, 2, 15, 4, 5, 87, 93, 96, 23	linecgrand 36046	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝑃⟩Cgr⟨𝑑, 𝑄⟩)
98	1, 2, 15, 26, 4, 5, 52, 61, 23, 97	linecgrand 36046	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑏, 𝑃⟩Cgr⟨𝑏, 𝑄⟩)
99		opeq1 4897	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ⟨𝑐, 𝑃⟩ = ⟨𝑏, 𝑃⟩)
100		opeq1 4897	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ⟨𝑐, 𝑄⟩ = ⟨𝑏, 𝑄⟩)
101	99, 100	breq12d 5179	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩ ↔ ⟨𝑏, 𝑃⟩Cgr⟨𝑏, 𝑄⟩))
102	101	biimprd 248	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (⟨𝑏, 𝑃⟩Cgr⟨𝑏, 𝑄⟩ → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩))
103	43, 98, 102	syl6ci 71	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐵 = 𝑏 → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩))
104	103	com12 32	. . . . . 6 ⊢ (𝐵 = 𝑏 → ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩))
105		simprl 770	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → 𝐵 ≠ 𝑏)
106	34	adantrl 715	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → 𝑐 Btwn ⟨𝐵, 𝑏⟩)
107		btwncolinear1 36033	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁))) → (𝑐 Btwn ⟨𝐵, 𝑏⟩ → 𝐵 Colinear ⟨𝑏, 𝑐⟩))
108	1, 25, 26, 3, 107	syl13anc 1372	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝑐 Btwn ⟨𝐵, 𝑏⟩ → 𝐵 Colinear ⟨𝑏, 𝑐⟩))
109	108	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → (𝑐 Btwn ⟨𝐵, 𝑏⟩ → 𝐵 Colinear ⟨𝑏, 𝑐⟩))
110	106, 109	mpd 15	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → 𝐵 Colinear ⟨𝑏, 𝑐⟩)
111		simp1rl 1238	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐵 Btwn ⟨𝐴, 𝐶⟩)
112	111	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐵 Btwn ⟨𝐴, 𝐶⟩)
113	1, 24, 25, 2, 15, 112, 54	btwnexch3and 35985	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Btwn ⟨𝐵, 𝑑⟩)
114		btwncolinear6 36038	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑑 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐵, 𝑑⟩ → 𝐶 Colinear ⟨𝑑, 𝐵⟩))
115	1, 25, 15, 2, 114	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐵, 𝑑⟩ → 𝐶 Colinear ⟨𝑑, 𝐵⟩))
116	115	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (𝐶 Btwn ⟨𝐵, 𝑑⟩ → 𝐶 Colinear ⟨𝑑, 𝐵⟩))
117	113, 116	mpd 15	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐶 Colinear ⟨𝑑, 𝐵⟩)
118	1, 2, 15, 25, 4, 5, 52, 117, 23, 97	linecgrand 36046	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝐵, 𝑃⟩Cgr⟨𝐵, 𝑄⟩)
119	118	adantrl 715	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → ⟨𝐵, 𝑃⟩Cgr⟨𝐵, 𝑄⟩)
120	98	adantrl 715	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → ⟨𝑏, 𝑃⟩Cgr⟨𝑏, 𝑄⟩)
121	1, 25, 26, 3, 4, 5, 105, 110, 119, 120	linecgrand 36046	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ (𝐵 ≠ 𝑏 ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩)
122	121	an12s 648	. . . . . . 7 ⊢ ((𝐵 ≠ 𝑏 ∧ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩)))))) → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩)
123	122	ex 412	. . . . . 6 ⊢ (𝐵 ≠ 𝑏 → ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩))
124	104, 123	pm2.61ine 3031	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑐, 𝑃⟩Cgr⟨𝑐, 𝑄⟩)
125	1, 2, 3, 4, 4, 5, 7, 13, 23, 124	linecgrand 36046	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑃, 𝑃⟩Cgr⟨𝑃, 𝑄⟩)
126		cgrid2 35967	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝑃⟩Cgr⟨𝑃, 𝑄⟩ → 𝑃 = 𝑄))
127	1, 4, 4, 5, 126	syl13anc 1372	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝑃⟩Cgr⟨𝑃, 𝑄⟩ → 𝑃 = 𝑄))
128	127	adantr 480	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → (⟨𝑃, 𝑃⟩Cgr⟨𝑃, 𝑄⟩ → 𝑃 = 𝑄))
129	125, 128	mpd 15	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑃 = 𝑄)
130		btwnconn1lem10 36060	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩)
131		opeq1 4897	. . . . . . 7 ⊢ (𝑃 = 𝑄 → ⟨𝑃, 𝑄⟩ = ⟨𝑄, 𝑄⟩)
132	131	breq2d 5178	. . . . . 6 ⊢ (𝑃 = 𝑄 → (⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩ ↔ ⟨𝑑, 𝐷⟩Cgr⟨𝑄, 𝑄⟩))
133	132	biimpa 476	. . . . 5 ⊢ ((𝑃 = 𝑄 ∧ ⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩) → ⟨𝑑, 𝐷⟩Cgr⟨𝑄, 𝑄⟩)
134		axcgrid 28949	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (⟨𝑑, 𝐷⟩Cgr⟨𝑄, 𝑄⟩ → 𝑑 = 𝐷))
135	1, 15, 14, 5, 134	syl13anc 1372	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (⟨𝑑, 𝐷⟩Cgr⟨𝑄, 𝑄⟩ → 𝑑 = 𝐷))
136	133, 135	syl5 34	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → ((𝑃 = 𝑄 ∧ ⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩) → 𝑑 = 𝐷))
137	136	adantr 480	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → ((𝑃 = 𝑄 ∧ ⟨𝑑, 𝐷⟩Cgr⟨𝑃, 𝑄⟩) → 𝑑 = 𝐷))
138	129, 130, 137	mp2and 698	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝑑 = 𝐷)
139	138	eqcomd 2746	1 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐷 = 𝑑)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  ℕcn 12293  𝔼cee 28921   Btwn cbtwn 28922  Cgrccgr 28923   Colinear ccolin 36001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-ee 28924  df-btwn 28925  df-cgr 28926  df-ofs 35947  df-colinear 36003  df-ifs 36004  df-cgr3 36005  df-fs 36006

This theorem is referenced by:  btwnconn1lem13  36063