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Proof step 48 from proof square-neq-any-pos-02 (previous step)

j / gcd(j, j0 * j1) * j / gcd(j0 * j1, j) = (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j) * i, j * j / gcd(j0 * j1, j) = (j0 * j1) * (j0 * j1) / gcd(j0 * j1, j) * i, gcd(j, j0) = 1, prime(j1), 1 < j1 * j1 * i, 0 ≤ j0, 0 < j, 0 < j0, 0 < j1, 0 < i, ∀ i1, j, j0. i→n(i1) < i→n(j1 * j1 * i) → 0 < i1 ∧ 0 < j ∧ 0 < j0 ∧ gcd(j, j0) = 1 ∧ j * j = i1 * j0 * j0 → (∃ i2. i1 = i2 * i2) ⊦ gcd(j0 * j1, j) = 1, j1 = 0, j = 0, ∃ i1. j1 * j1 * i = i1 * i1

User applied rule apply induction

And obtained the following new goals

Proof step 49:
j / gcd(j, j0 * j1) * j / gcd(j0 * j1, j) = (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j) * i, j * j / gcd(j0 * j1, j) = (j0 * j1) * (j0 * j1) / gcd(j0 * j1, j) * i, gcd(j, j0) = 1, prime(j1), 1 < j1 * j1 * i, 0 ≤ j0, 0 < j, 0 < j0, 0 < j1, 0 < i, ∀ i1, j, j0. i→n(i1) < i→n(j1 * j1 * i) → 0 < i1 ∧ 0 < j ∧ 0 < j0 ∧ gcd(j, j0) = 1 ∧ j * j = i1 * j0 * j0 → (∃ i2. i1 = i2 * i2) ⊦ i→n(i) < i→n(j1 * j1 * i), gcd(j0 * j1, j) = 1, j1 = 0, j = 0, ∃ i1. j1 * j1 * i = i1 * i1
Proof step 52:
j / gcd(j, j0 * j1) * j / gcd(j0 * j1, j) = (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j) * i, j * j / gcd(j0 * j1, j) = (j0 * j1) * (j0 * j1) / gcd(j0 * j1, j) * i, gcd(j, j0) = 1, prime(j1), 1 < j1 * j1 * i, 0 ≤ j0, 0 < j, 0 < j0, 0 < j1, 0 < i, ∀ i1, j, j0. i→n(i1) < i→n(j1 * j1 * i) → 0 < i1 ∧ 0 < j ∧ 0 < j0 ∧ gcd(j, j0) = 1 ∧ j * j = i1 * j0 * j0 → (∃ i2. i1 = i2 * i2) ⊦ 0 < i ∧ 0 < j / gcd(j, j0 * j1) ∧ 0 < (j0 * j1) / gcd(j0 * j1, j) ∧ gcd(j / gcd(j, j0 * j1), (j0 * j1) / gcd(j0 * j1, j)) = 1 ∧ j / gcd(j, j0 * j1) * j / gcd(j, j0 * j1) = i * (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j), gcd(j0 * j1, j) = 1, j1 = 0, j = 0, ∃ i1. j1 * j1 * i = i1 * i1
Proof step 57:
∃ i2. i = i2 * i2, j / gcd(j, j0 * j1) * j / gcd(j0 * j1, j) = (j0 * j1) / gcd(j0 * j1, j) * (j0 * j1) / gcd(j0 * j1, j) * i, j * j / gcd(j0 * j1, j) = (j0 * j1) * (j0 * j1) / gcd(j0 * j1, j) * i, gcd(j, j0) = 1, prime(j1), 1 < j1 * j1 * i, 0 ≤ j0, 0 < j, 0 < j0, 0 < j1, 0 < i, ∀ i1, j, j0. i→n(i1) < i→n(j1 * j1 * i) → 0 < i1 ∧ 0 < j ∧ 0 < j0 ∧ gcd(j, j0) = 1 ∧ j * j = i1 * j0 * j0 → (∃ i2. i1 = i2 * i2) ⊦ gcd(j0 * j1, j) = 1, j1 = 0, j = 0, ∃ i1. j1 * j1 * i = i1 * i1

Rule argument

subst: [i1, j, j0] → [i, j / gcd(j, j0 * j1), (j0 * j1) / gcd(j0 * j1, j)]

Additional information

Used terms: i, j / gcd(j, j0 * j1), (j0 * j1) / gcd(j0 * j1, j)