Metamath Proof Explorer


Theorem btwnconn1lem6

Description: Lemma for btwnconn1 . Next, we show that E is the midpoint of D and d . (Contributed by Scott Fenton, 8-Oct-2013)

Ref Expression
Assertion btwnconn1lem6 ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ )

Proof

Step Hyp Ref Expression
1 simprrl ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 Btwn ⟨ 𝐶 , 𝑐 ⟩ )
2 1 1 jca ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 Btwn ⟨ 𝐶 , 𝑐 ⟩ ∧ 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ) )
3 simp11 ( ( ( 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 ) ) ) → 𝑁 ∈ ℕ )
4 simp21 ( ( ( 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 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
5 simp23 ( ( ( 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 ) ) ) → 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) )
6 3 4 5 cgrrflxd ( ( ( 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 ) ) ) → ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ )
7 simp33 ( ( ( 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 ) ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
8 3 7 5 cgrrflxd ( ( ( 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 ) ) ) → ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ )
9 6 8 jca ( ( ( 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 ) ) ) → ( ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) )
10 9 adantr ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) )
11 simp31 ( ( ( 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 ) ) ) → 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) )
12 simp22 ( ( ( 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 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
13 simp2rr ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ )
14 13 ad2antrl ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
15 3 4 11 4 12 14 cgrcomand ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ )
16 simp2lr ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
17 16 ad2antrl ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
18 3 12 5 4 12 17 cgrcomrand ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ )
19 3simpa ( ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) )
20 19 3anim3i ( ( ( 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 ) ) ) → ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) )
21 simpl ( ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) → ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) ) )
22 btwnconn1lem4 ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ )
23 20 21 22 syl2an ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ )
24 3 12 5 11 5 12 4 18 23 cgrtr3and ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝑑 , 𝑐 ⟩ )
25 3 12 5 11 5 24 cgrcomlrand ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ )
26 15 25 jca ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ ) )
27 brifs ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐶 , 𝐸 ⟩ , ⟨ 𝑐 , 𝐷 ⟩ ⟩ InnerFiveSeg ⟨ ⟨ 𝐶 , 𝐸 ⟩ , ⟨ 𝑐 , 𝑑 ⟩ ⟩ ↔ ( ( 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ∧ 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ ) ) ) )
28 ifscgr ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐶 , 𝐸 ⟩ , ⟨ 𝑐 , 𝐷 ⟩ ⟩ InnerFiveSeg ⟨ ⟨ 𝐶 , 𝐸 ⟩ , ⟨ 𝑐 , 𝑑 ⟩ ⟩ → ⟨ 𝐸 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ ) )
29 27 28 sylbird ( ( ( 𝑁 ∈ ℕ ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ∧ 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ ) ) → ⟨ 𝐸 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ ) )
30 3 4 7 5 12 4 7 5 11 29 syl333anc ( ( ( 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 ) ) ) → ( ( ( 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ∧ 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ ) ) → ⟨ 𝐸 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ ) )
31 30 adantr ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 Btwn ⟨ 𝐶 , 𝑐 ⟩ ∧ 𝐸 Btwn ⟨ 𝐶 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝑐 ⟩ ∧ ⟨ 𝐸 , 𝑐 ⟩ Cgr ⟨ 𝐸 , 𝑐 ⟩ ) ∧ ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝑐 , 𝑑 ⟩ ) ) → ⟨ 𝐸 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ ) )
32 2 10 26 31 mp3and ( ( ( ( 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 ) ) ) ∧ ( ( ( ( 𝐴𝐵𝐵𝐶𝐶𝑐 ) ∧ ( 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 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝑑 ⟩ )