Metamath Proof Explorer

Theorem btwnconn1lem1

Description: Lemma for btwnconn1 . The next several lemmas introduce various properties of hypothetical points that end up eliminating alternatives to connectivity. We begin by showing a congruence property of those hypothetical points. (Contributed by Scott Fenton, 8-Oct-2013)

Ref Expression
Assertion btwnconn1lem1 ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑋 , 𝐶 ⟩ )

Proof

Step Hyp Ref Expression
1 simp11 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
2 simp13 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
3 simp22 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
4 simp23 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) )
5 simp33 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) )
6 simp31 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) )
7 simp21 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
8 simp12 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
9 simp1rr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ )
10 9 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ )
11 simp2ll ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ )
12 11 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ )
13 1 8 2 3 4 10 12 btwnexch3and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐷 Btwn ⟨ 𝐵 , 𝑐 ⟩ )
14 simp2rl ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ )
15 14 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ )
16 simp3rl ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝑑 Btwn ⟨ 𝐴 , 𝑋 ⟩ )
17 16 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝑑 Btwn ⟨ 𝐴 , 𝑋 ⟩ )
18 1 8 7 6 5 15 17 btwnexch3and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝑑 Btwn ⟨ 𝐶 , 𝑋 ⟩ )
19 1 6 7 5 18 btwncomand ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝑑 Btwn ⟨ 𝑋 , 𝐶 ⟩ )
20 simp3rr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ )
21 20 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ )
22 1 6 5 3 2 21 cgrcomand ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐷 , 𝐵 ⟩ Cgr ⟨ 𝑑 , 𝑋 ⟩ )
23 1 3 2 6 5 22 cgrcomlrand ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝑋 , 𝑑 ⟩ )
24 simp2lr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
25 24 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
26 simp2rr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
27 26 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
28 1 7 6 7 3 27 cgrcomland ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑑 , 𝐶 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
29 1 3 4 6 7 7 3 25 28 cgrtr3and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝑑 , 𝐶 ⟩ )
30 1 2 3 4 5 6 7 13 19 23 29 cgrextendand ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑋 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 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 , 𝑋 ⟩ ∧ ⟨ 𝑑 , 𝑋 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑋 , 𝐶 ⟩ )