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Proof step 73 from proof Lagrange-SumOf4Squares-primes (previous step)

j0 ^2 * j * i7 = j0 ^2 * (i11 ^2 + i10 ^2 + i8 ^2 + i9 ^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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 74:
i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, j * i7 = i11 ^2 + i10 ^2 + i8 ^2 + i9 ^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

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%eq-pos: ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); from specification int-mult