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Proof step 6 from proof euler-four-square-identity (previous step)

⊦ (i1 ^2 * j1 ^2 + (i2 ^2 + i3 ^2 + i4 ^2) * j1 ^2) + (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^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 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 8:
⊦ (i1 ^2 * j1 ^2 + i2 ^2 * j1 ^2 + (i3 ^2 + i4 ^2) * j1 ^2) + (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^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 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2

Rule argument

spec: int-mult name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [j1 ^2, i3 ^2 + i4 ^2, i2 ^2] flags: <all paths>

Additional information

Used terms: j1 ^2, i3 ^2 + i4 ^2, i2 ^2