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Proof step 46 from proof Lagrange-SumOf4Squares-primes (previous step)

i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, i7 < j, 0 < j0, 1 < j0, 0 < i7, j0 * i7 < j0 ^2, 0 < j, prime(j), j0 < j, abs(i3) ≤ j0 / 2, ≡mod(i3, i2, j0), abs(i4) ≤ j0 / 2, ≡mod(i4, i1, j0), abs(i5) ≤ j0 / 2, ≡mod(i5, i0, j0), abs(i6) ≤ j0 / 2, ≡mod(i6, i, j0), 1 < j0 ^2, 0 < j0 ^2, i7 < j0, 1 < j, ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ j = 2, j0 = 1, even(j0)

User applied rule cut formula

And obtained the following new goals

Proof step 47:
j0 * j * j0 * i7 = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) * (i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2), i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, i7 < j, 0 < j0, 1 < j0, 0 < i7, j0 * i7 < j0 ^2, 0 < j, prime(j), j0 < j, abs(i3) ≤ j0 / 2, ≡mod(i3, i2, j0), abs(i4) ≤ j0 / 2, ≡mod(i4, i1, j0), abs(i5) ≤ j0 / 2, ≡mod(i5, i0, j0), abs(i6) ≤ j0 / 2, ≡mod(i6, i, j0), 1 < j0 ^2, 0 < j0 ^2, i7 < j0, 1 < j, ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ j = 2, j0 = 1, even(j0)

Rule argument

fma: j0 * j * j0 * i7 = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) * (i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2)

Additional information

Used simplifier rules: neq-not-zero: ⊦ i * j ≠ i * j0 → i ≠ 0; (forward) from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: j0 * j * j0 * i7 = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) * (i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2)