Theorem axcontlem8

Description: Lemma for axcont . A point in D is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013)

	Ref	Expression
Hypotheses	axcontlem8.1	⊢ 𝐷 = { 𝑝 ∈ ( 𝔼 ‘ 𝑁 ) ∣ ( 𝑈 Btwn ⟨ 𝑍 , 𝑝 ⟩ ∨ 𝑝 Btwn ⟨ 𝑍 , 𝑈 ⟩ ) }
	axcontlem8.2	⊢ 𝐹 = { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐷 ∧ ( 𝑡 ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑥 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑈 ‘ 𝑖 ) ) ) ) ) }
Assertion	axcontlem8	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) → 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		axcontlem8.1	⊢ 𝐷 = { 𝑝 ∈ ( 𝔼 ‘ 𝑁 ) ∣ ( 𝑈 Btwn ⟨ 𝑍 , 𝑝 ⟩ ∨ 𝑝 Btwn ⟨ 𝑍 , 𝑈 ⟩ ) }
2		axcontlem8.2	⊢ 𝐹 = { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐷 ∧ ( 𝑡 ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑥 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑍 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑈 ‘ 𝑖 ) ) ) ) ) }
3	1 2	axcontlem6	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ 𝑃 ∈ 𝐷 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
4	3	ex	⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → ( 𝑃 ∈ 𝐷 → ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
5	1 2	axcontlem6	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ 𝑄 ∈ 𝐷 ) → ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
6	5	ex	⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → ( 𝑄 ∈ 𝐷 → ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
7	1 2	axcontlem6	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ 𝑅 ∈ 𝐷 ) → ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
8	7	ex	⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → ( 𝑅 ∈ 𝐷 → ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
9	4 6 8	3anim123d	⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → ( ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
10	9	imp	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
11	10	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
12		3an6	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ∧ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
13		0elunit	⊢ 0 ∈ ( 0 [,] 1 )
14		simplr1	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) )
15	14	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) )
16		elrege0	⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑃 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑃 ) ) )
17	16	simplbi	⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ )
18	15 17	syl	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ )
19	18	recnd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ )
20		simprrl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) )
21	20	adantr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) )
22		simprrr	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) )
23		simpl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) )
24	22 23	breqtrrd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑃 ) )
25	24	adantr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑃 ) )
26		simplr2	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) )
27	26	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) )
28		elrege0	⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑄 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑄 ) ) )
29	28	simplbi	⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ )
30	27 29	syl	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ )
31	18 30	letri3d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ↔ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑃 ) ) ) )
32	21 25 31	mpbir2and	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) )
33		simpll	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) )
34		simpll2	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) )
35	34	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) )
36	35	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) )
37		fveecn	⊢ ( ( 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ )
38	36 37	sylancom	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ )
39		simpll3	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) → 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) )
40	39	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) )
41	40	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) )
42		fveecn	⊢ ( ( 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ )
43	41 42	sylancom	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ )
44		ax-1cn	⊢ 1 ∈ ℂ
45		simpl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ )
46		subcl	⊢ ( ( 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
47	44 45 46	sylancr	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
48		simprl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ )
49	47 48	mulcld	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ )
50		mulcl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ )
51	50	adantrl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ )
52	49 51	addcld	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∈ ℂ )
53	52	mullidd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
54	52	mul02d	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = 0 )
55	53 54	oveq12d	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + 0 ) )
56	52	addridd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + 0 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
57	55 56	eqtr2d	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
58	57	3adant2	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
59		oveq2	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) = ( 1 − ( 𝐹 ‘ 𝑄 ) ) )
60	59	oveq1d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) → ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) )
61		oveq1	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) )
62	60 61	oveq12d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) → ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
63		oveq2	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) = ( 1 − ( 𝐹 ‘ 𝑅 ) ) )
64	63	oveq1d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) )
65		oveq1	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) )
66	64 65	oveq12d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
67	66	oveq2d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
68	67	oveq2d	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
69	62 68	eqeqan12d	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
70	69	3ad2ant2	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
71	58 70	mpbid	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
72	19 32 33 38 43 71	syl122anc	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
73	72	ralrimiva	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
74		oveq2	⊢ ( 𝑡 = 0 → ( 1 − 𝑡 ) = ( 1 − 0 ) )
75		1m0e1	⊢ ( 1 − 0 ) = 1
76	74 75	eqtrdi	⊢ ( 𝑡 = 0 → ( 1 − 𝑡 ) = 1 )
77	76	oveq1d	⊢ ( 𝑡 = 0 → ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
78		oveq1	⊢ ( 𝑡 = 0 → ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
79	77 78	oveq12d	⊢ ( 𝑡 = 0 → ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
80	79	eqeq2d	⊢ ( 𝑡 = 0 → ( ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
81	80	ralbidv	⊢ ( 𝑡 = 0 → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
82	81	rspcev	⊢ ( ( 0 ∈ ( 0 [,] 1 ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( 1 · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 0 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
83	13 73 82	sylancr	⊢ ( ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
84	83	ex	⊢ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑅 ) → ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
85	26	adantl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) )
86	85 29	syl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℝ )
87		simplr3	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) )
88	87	adantl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) )
89		elrege0	⊢ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑅 ) ∈ ℝ ∧ 0 ≤ ( 𝐹 ‘ 𝑅 ) ) )
90	89	simplbi	⊢ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑅 ) ∈ ℝ )
91	88 90	syl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ℝ )
92	14	adantl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) )
93	92 17	syl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℝ )
94		simprrr	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) )
95	86 91 93 94	lesub1dd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ≤ ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) )
96	86 93	resubcld	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℝ )
97		simprrl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) )
98	86 93	subge0d	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 0 ≤ ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ↔ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ) )
99	97 98	mpbird	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → 0 ≤ ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) )
100	91 93	resubcld	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℝ )
101	93 86 91 97 94	letrd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑅 ) )
102		simpl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) )
103	102	necomd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) ≠ ( 𝐹 ‘ 𝑃 ) )
104	93 91 101 103	leneltd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) < ( 𝐹 ‘ 𝑅 ) )
105	93 91	posdifd	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) < ( 𝐹 ‘ 𝑅 ) ↔ 0 < ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) )
106	104 105	mpbid	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → 0 < ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) )
107		divelunit	⊢ ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℝ ∧ 0 ≤ ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℝ ∧ 0 < ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ≤ ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) )
108	96 99 100 106 107	syl22anc	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ( 0 [,] 1 ) ↔ ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ≤ ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) )
109	95 108	mpbird	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ( 0 [,] 1 ) )
110	14	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) )
111	17	recnd	⊢ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ )
112	110 111	syl	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ )
113		simpll	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) )
114	26	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) )
115	29	recnd	⊢ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ )
116	114 115	syl	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ )
117	87	ad2antlr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) )
118	90	recnd	⊢ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) → ( 𝐹 ‘ 𝑅 ) ∈ ℂ )
119	117 118	syl	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ℂ )
120	34	ad2antrl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) )
121	120 37	sylan	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ )
122	39	ad2antrl	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) )
123	122 42	sylan	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ )
124		simp2r	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑅 ) ∈ ℂ )
125		simp2l	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ℂ )
126	124 125	subcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ )
127		simp1l	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ℂ )
128	44 127 46	sylancr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
129	126 128	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ℂ )
130	125 127	subcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
131		subcl	⊢ ( ( 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
132	44 124 131	sylancr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
133	130 132	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ∈ ℂ )
134	124 127	subcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
135		simp1r	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) )
136	135	necomd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑅 ) ≠ ( 𝐹 ‘ 𝑃 ) )
137	124 127 136	subne0d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ≠ 0 )
138	129 133 134 137	divdird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
139	134	mulridd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) = ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) )
140	134 125	mulcomd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) · ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) )
141	125 124 127	subdid	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) · ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
142	140 141	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) = ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
143	139 142	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) )
144		subdi	⊢ ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) )
145	44 144	mp3an2	⊢ ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) )
146	134 125 145	syl2anc	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) ) )
147		subdi	⊢ ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) ) )
148	44 147	mp3an2	⊢ ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) ) )
149	126 127 148	syl2anc	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) ) )
150	126	mulridd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · 1 ) = ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) )
151	124 125 127	subdird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝐹 ‘ 𝑅 ) · ( 𝐹 ‘ 𝑃 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
152	124 127	mulcomd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑅 ) · ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) )
153	152	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) · ( 𝐹 ‘ 𝑃 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
154	151 153	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
155	150 154	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) − ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) )
156	149 155	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) − ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) )
157		subdi	⊢ ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
158	44 157	mp3an2	⊢ ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
159	130 124 158	syl2anc	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
160	130	mulridd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) = ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) )
161	125 127 124	subdird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) = ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) )
162	160 161	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · 1 ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) )
163	159 162	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) )
164	156 163	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) − ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) ) )
165	127 124	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
166	125 127	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
167	165 166	subcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ∈ ℂ )
168		mulcl	⊢ ( ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) → ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
169	168	3ad2ant2	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
170	169 165	subcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ∈ ℂ )
171	126 130 167 170	addsub4d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) + ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) − ( ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) − ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) ) )
172	124 125 127	npncand	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) + ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) )
173	165 166 169	npncan3d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) = ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) )
174	172 173	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) + ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) − ( ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) )
175	164 171 174	3eqtr2d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) ) )
176	143 146 175	3eqtr4d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) )
177	129 133	addcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) ∈ ℂ )
178		subcl	⊢ ( ( 1 ∈ ℂ ∧ ( 𝐹 ‘ 𝑄 ) ∈ ℂ ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ )
179	44 125 178	sylancr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) ∈ ℂ )
180	177 134 179 137	divmuld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( 1 − ( 𝐹 ‘ 𝑄 ) ) ↔ ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑄 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) ) )
181	176 180	mpbird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( 1 − ( 𝐹 ‘ 𝑄 ) ) )
182	126 128 134 137	div23d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) )
183	134 130 134 137	divsubdird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
184	124 125 127	nnncan2d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) )
185	184	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) − ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) )
186	134 137	dividd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = 1 )
187	186	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) = ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
188	183 185 187	3eqtr3d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
189	188	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) )
190	182 189	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) )
191	130 132 134 137	div23d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) )
192	190 191	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) )
193	138 181 192	3eqtr3d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( 𝐹 ‘ 𝑄 ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) )
194	193	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) )
195	126 127	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
196	130 124	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
197	195 196 134 137	divdird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
198	154 161	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( 𝐹 ‘ 𝑄 ) · ( 𝐹 ‘ 𝑅 ) ) − ( ( 𝐹 ‘ 𝑃 ) · ( 𝐹 ‘ 𝑅 ) ) ) ) )
199	173 198 142	3eqtr4rd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
200	195 196	addcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) ∈ ℂ )
201	200 134 125 137	divmuld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( 𝐹 ‘ 𝑄 ) ↔ ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑄 ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) ) )
202	199 201	mpbird	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( 𝐹 ‘ 𝑄 ) )
203	126 127 134 137	div23d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑃 ) ) )
204	188	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑃 ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) )
205	203 204	eqtrd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) )
206	130 124 134 137	div23d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) )
207	205 206	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑄 ) ) · ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) · ( 𝐹 ‘ 𝑅 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
208	197 202 207	3eqtr3d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) )
209	208	oveq1d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) )
210	194 209	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) )
211	130 134 137	divcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ℂ )
212		subcl	⊢ ( ( 1 ∈ ℂ ∧ ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ℂ ) → ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) ∈ ℂ )
213	44 211 212	sylancr	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) ∈ ℂ )
214		simp3l	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝑍 ‘ 𝑖 ) ∈ ℂ )
215	128 214	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ )
216	213 215	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ∈ ℂ )
217	132 214	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ∈ ℂ )
218	211 217	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ∈ ℂ )
219		simp3r	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( 𝑈 ‘ 𝑖 ) ∈ ℂ )
220	127 219	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ )
221	213 220	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∈ ℂ )
222	124 219	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ∈ ℂ )
223	211 222	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∈ ℂ )
224	216 218 221 223	add4d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
225	213 128	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ℂ )
226	211 132	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ∈ ℂ )
227	213 128 214	mulassd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) )
228	211 132 214	mulassd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) )
229	227 228	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑍 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) )
230	225 214 226 229	joinlmuladdmuld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) )
231	213 127	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) ∈ ℂ )
232	211 124	mulcld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ∈ ℂ )
233	213 127 219	mulassd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
234	211 124 219	mulassd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
235	233 234	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) · ( 𝑈 ‘ 𝑖 ) ) + ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
236	231 219 232 235	joinlmuladdmuld	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
237	230 236	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) ) + ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
238	213 215 220	adddid	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
239	211 217 222	adddid	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
240	238 239	oveq12d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
241	224 237 240	3eqtr4rd	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 1 − ( 𝐹 ‘ 𝑃 ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 1 − ( 𝐹 ‘ 𝑅 ) ) ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( 𝐹 ‘ 𝑃 ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( 𝐹 ‘ 𝑅 ) ) ) · ( 𝑈 ‘ 𝑖 ) ) ) )
242	210 241	eqtr4d	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ℂ ∧ ( 𝐹 ‘ 𝑅 ) ∈ ℂ ) ∧ ( ( 𝑍 ‘ 𝑖 ) ∈ ℂ ∧ ( 𝑈 ‘ 𝑖 ) ∈ ℂ ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
243	112 113 116 119 121 123 242	syl222anc	⊢ ( ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
244	243	ralrimiva	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
245		oveq2	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( 1 − 𝑡 ) = ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) )
246	245	oveq1d	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
247		oveq1	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) = ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
248	246 247	oveq12d	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
249	248	eqeq2d	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
250	249	ralbidv	⊢ ( 𝑡 = ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
251	250	rspcev	⊢ ( ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ∈ ( 0 [,] 1 ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( ( ( ( 𝐹 ‘ 𝑄 ) − ( 𝐹 ‘ 𝑃 ) ) / ( ( 𝐹 ‘ 𝑅 ) − ( 𝐹 ‘ 𝑃 ) ) ) · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
252	109 244 251	syl2anc	⊢ ( ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) ∧ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
253	252	ex	⊢ ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑅 ) → ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
254	84 253	pm2.61ine	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
255		r19.26-3	⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ↔ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
256		simp2	⊢ ( ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) )
257		oveq2	⊢ ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) → ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) = ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
258		oveq2	⊢ ( ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) → ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) = ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) )
259	257 258	oveqan12d	⊢ ( ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
260	259	3adant2	⊢ ( ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) )
261	256 260	eqeq12d	⊢ ( ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
262	261	ralimi	⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
263		ralbi	⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
264	262 263	syl	⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
265	264	rexbidv	⊢ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) ) )
266	265	biimprcd	⊢ ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
267	255 266	biimtrrid	⊢ ( ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) = ( ( ( 1 − 𝑡 ) · ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) + ( 𝑡 · ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ( ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
268	254 267	syl	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
269	268	an32s	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ) → ( ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
270	269	expimpd	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ∧ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
271	270	adantlr	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ) ∧ ( ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
272	12 271	biimtrid	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( ( ( ( 𝐹 ‘ 𝑃 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑃 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑃 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑃 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑄 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑄 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ∧ ( ( 𝐹 ‘ 𝑅 ) ∈ ( 0 [,) +∞ ) ∧ ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑅 ‘ 𝑖 ) = ( ( ( 1 − ( 𝐹 ‘ 𝑅 ) ) · ( 𝑍 ‘ 𝑖 ) ) + ( ( 𝐹 ‘ 𝑅 ) · ( 𝑈 ‘ 𝑖 ) ) ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
273	11 272	mpd	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) )
274		simpl1	⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) → 𝑁 ∈ ℕ )
275	1	ssrab3	⊢ 𝐷 ⊆ ( 𝔼 ‘ 𝑁 )
276	275	sseli	⊢ ( 𝑄 ∈ 𝐷 → 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) )
277	275	sseli	⊢ ( 𝑃 ∈ 𝐷 → 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) )
278	275	sseli	⊢ ( 𝑅 ∈ 𝐷 → 𝑅 ∈ ( 𝔼 ‘ 𝑁 ) )
279	276 277 278	3anim123i	⊢ ( ( 𝑄 ∈ 𝐷 ∧ 𝑃 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) → ( 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑅 ∈ ( 𝔼 ‘ 𝑁 ) ) )
280	279	3com12	⊢ ( ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) → ( 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑅 ∈ ( 𝔼 ‘ 𝑁 ) ) )
281		brbtwn	⊢ ( ( 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑅 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
282	281	adantl	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑄 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑃 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑅 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
283	274 280 282	syl2an	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) → ( 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
284	283	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → ( 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ↔ ∃ 𝑡 ∈ ( 0 [,] 1 ) ∀ 𝑖 ∈ ( 1 ... 𝑁 ) ( 𝑄 ‘ 𝑖 ) = ( ( ( 1 − 𝑡 ) · ( 𝑃 ‘ 𝑖 ) ) + ( 𝑡 · ( 𝑅 ‘ 𝑖 ) ) ) ) )
285	273 284	mpbird	⊢ ( ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) ) → 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ )
286	285	ex	⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝑍 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑈 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ 𝑍 ≠ 𝑈 ) ∧ ( 𝑃 ∈ 𝐷 ∧ 𝑄 ∈ 𝐷 ∧ 𝑅 ∈ 𝐷 ) ) → ( ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝐹 ‘ 𝑄 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( 𝐹 ‘ 𝑅 ) ) → 𝑄 Btwn ⟨ 𝑃 , 𝑅 ⟩ ) )