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Proof step 77 from proof Extsimple (previous step)

abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2), abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2), (0 < i1 ∧ 0 < j1 ∨ i1 < 0 ∧ j1 < 0) ∧ (0 < i2 ∧ 0 < j2 ∨ i2 < 0 ∧ j2 < 0) ∨ (0 < i1 ∧ j1 < 0 ∨ i1 < 0 ∧ 0 < j1) ∧ (0 < i2 ∧ j2 < 0 ∨ i2 < 0 ∧ 0 < j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, rational(i1, j1).num = rational(i2, j2).num

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 78:
⊦ i ≠ 0 ∧ 0 < j → rational(i, j).num = i / gcd(i, j)
Proof step 79:
¬(i1 ≠ 0 ∧ 0 < j1), abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2), abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2), (0 < i1 ∧ 0 < j1 ∨ i1 < 0 ∧ j1 < 0) ∧ (0 < i2 ∧ 0 < j2 ∨ i2 < 0 ∧ j2 < 0) ∨ (0 < i1 ∧ j1 < 0 ∨ i1 < 0 ∧ 0 < j1) ∧ (0 < i2 ∧ j2 < 0 ∨ i2 < 0 ∧ 0 < j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, rational(i1, j1).num = rational(i2, j2).num
Proof step 88:
i1 ≠ 0 ∧ 0 < j1, abs(i1) / gcd(i1, j1) = abs(i2) / gcd(i2, j2), abs(j1) / gcd(i1, j1) = abs(j2) / gcd(i2, j2), (0 < i1 ∧ 0 < j1 ∨ i1 < 0 ∧ j1 < 0) ∧ (0 < i2 ∧ 0 < j2 ∨ i2 < 0 ∧ j2 < 0) ∨ (0 < i1 ∧ j1 < 0 ∨ i1 < 0 ∧ 0 < j1) ∧ (0 < i2 ∧ j2 < 0 ∨ i2 < 0 ∧ 0 < j2) ⊦ j2 = 0, j1 = 0, i2 = 0, i1 = 0, i1 / gcd(i1, j1) = rational(i2, j2).num

Rule argument

name: numerator-pos rotate: F subst: [j, i] → [j1, i1]

Additional information

Used terms: j1, i1