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Proof step 27 from proof Lagrange-SumOf4Squares-primes (previous step)

(i4 + i3) % 2 = 0, (i6 + i5) % 2 = 0, 2 * (((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2) + 2 * (((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^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 28:
⊦ i * j + i * j0 = i * (j + j0)

Rule argument

spec: int-mult inst: name: dist-rmerge seq: ⊦ i * j + i * j0 = i * (j + j0) subst: [j0, j, i] → [((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^2, ((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2, 2] flags: rewrite

Additional information

Used simplifier rules: eq-pos: ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: ((i4 - i3) / 2)^2 + ((i4 + i3) / 2)^2, ((i6 - i5) / 2)^2 + ((i6 + i5) / 2)^2, 2