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Proof step 21 from proof Lagrange-SumOf4Squares-primes (previous step)

(i6 ^2 + i5 ^2) + i4 ^2 + i3 ^2 = (2 * j1) * j, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = (2 * j1) * j, prime(j), 0 < j, 1 < j, j1 < j, even(i6 + i5), even(i4 + i3), 2 * j1 < j, 0 < j1 ⊦ j = 2, ((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^2 + ((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2 = j1 * j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ (j1 + j2) % 2 = 0 → j1 ^2 + j2 ^2 = 2 * (((j1 - j2) / 2)^2 + ((j1 + j2) / 2)^2)
Proof step 23:
(i6 + i5) % 2 ≠ 0, (i6 ^2 + i5 ^2) + i4 ^2 + i3 ^2 = (2 * j1) * j, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = (2 * j1) * j, prime(j), 0 < j, 1 < j, j1 < j, even(i6 + i5), even(i4 + i3), 2 * j1 < j, 0 < j1 ⊦ j = 2, ((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^2 + ((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2 = j1 * j
Proof step 24:
(i6 + i5) % 2 = 0, 2 * (((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2) + i4 ^2 + i3 ^2 = (2 * j1) * j, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = (2 * j1) * j, prime(j), 0 < j, 1 < j, j1 < j, even(i6 + i5), even(i4 + i3), 2 * j1 < j, 0 < j1 ⊦ j = 2, ((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^2 + ((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2 = j1 * j

Rule argument

spec: int-square inst: name: two-even-01 seq: ⊦ (j1 + j2) % 2 = 0 → j1 ^2 + j2 ^2 = 2 * (((j1 - j2) / 2)^2 + ((j1 + j2) / 2)^2) subst: [j2, j1] → [i5, i6] flags: rewrite

Additional information

Used terms: i5, i6