Step
*
1
of Lemma
rpositive2_functionality
1. x : ℕ+ ⟶ ℤ@i
2. y : ℕ+ ⟶ ℤ@i
3. bdd-diff(x;y)@i
4. n : ℕ+@i
5. ∀m:ℕ+. ((n ≤ m)
⇒ (m ≤ (n * (x m))))@i
⊢ ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m)
⇒ (m ≤ (n * (y m))))
BY
{ (D 3 THEN ((With ⌜(B + 1) * n⌝ (D 0)⋅ THENA Auto) THEN RepeatFor 2 (ParallelLast) THEN Auto')⋅) }
1
.....antecedent.....
1. x : ℕ+ ⟶ ℤ@i
2. y : ℕ+ ⟶ ℤ@i
3. B : ℕ@i
4. ∀n:ℕ+. (|(x n) - y n| ≤ B)@i
5. n : ℕ+@i
6. ∀m:ℕ+. ((n ≤ m)
⇒ (m ≤ (n * (x m))))@i
7. m : ℕ+@i
8. ((B + 1) * n) ≤ m@i
⊢ n ≤ m
2
1. x : ℕ+ ⟶ ℤ@i
2. y : ℕ+ ⟶ ℤ@i
3. B : ℕ@i
4. ∀n:ℕ+. (|(x n) - y n| ≤ B)@i
5. n : ℕ+@i
6. ∀m:ℕ+. ((n ≤ m)
⇒ (m ≤ (n * (x m))))@i
7. m : ℕ+@i
8. ((B + 1) * n) ≤ m@i
9. m ≤ (n * (x m))
⊢ m ≤ (((B + 1) * n) * (y m))
Latex:
Latex:
1. x : \mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}@i
2. y : \mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}@i
3. bdd-diff(x;y)@i
4. n : \mBbbN{}\msupplus{}@i
5. \mforall{}m:\mBbbN{}\msupplus{}. ((n \mleq{} m) {}\mRightarrow{} (m \mleq{} (n * (x m))))@i
\mvdash{} \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\mBbbN{}\msupplus{}. ((n \mleq{} m) {}\mRightarrow{} (m \mleq{} (n * (y m))))
By
Latex:
(D 3 THEN ((With \mkleeneopen{}(B + 1) * n\mkleeneclose{} (D 0)\mcdot{} THENA Auto) THEN RepeatFor 2 (ParallelLast) THEN Auto')\mcdot{})
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