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Proof step 47 from proof Lagrange-SumOf4Squares-primes (previous step)

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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^2) * (j1 ^2 + j2 ^2 + j3 ^2 + j4 ^2) = (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + ((i1 * j2 - i2 * j1 - i3 * j4) + i4 * j3)^2 + (i1 * j3 + i2 * j4 - i3 * j1 - i4 * j2)^2 + ((i1 * j4 - i2 * j3) + i3 * j2 - i4 * j1)^2
Proof step 49:
j0 * j * j0 * i7 = (((i2 * i3 + i1 * i4) + i0 * i5) + i * i6)^2 + ((i2 * i4 - i1 * i3 - i0 * i6) + i * i5)^2 + (i2 * i5 + i1 * i6 - i0 * i3 - i * i4)^2 + ((i2 * i6 - i1 * i5) + i0 * i4 - i * i3)^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

spec: int-square inst: name: euler-four-square-identity2 seq: ⊦ (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^2) * (j1 ^2 + j2 ^2 + j3 ^2 + j4 ^2) = (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + ((i1 * j2 - i2 * j1 - i3 * j4) + i4 * j3)^2 + (i1 * j3 + i2 * j4 - i3 * j1 - i4 * j2)^2 + ((i1 * j4 - i2 * j3) + i3 * j2 - i4 * j1)^2 subst: [j4, j3, j2, j1, i4, i3, i2, i1] → [i6, i5, i4, i3, i, i0, i1, i2] flags: rewrite

Additional information

Used terms: i6, i5, i4, i3, i, i0, i1, i2