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Proof step 40 from proof euler-four-square-identity2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 41:
⊦ i * (j + j0) = i * j + i * j0
Proof step 42:
⊦ i4 ^2 * j3 ^2 + 2 * i4 * j3 * i1 * j2 + (i2 * j3 + i4 * j1)^2 + (i3 * j1 + i4 * j2)^2 + (i2 * j1 + i3 * j4)^2 + (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + (i3 * j2 + i1 * j4)^2 + (i1 * j3 + i2 * j4)^2 = i1 ^2 * j1 ^2 + i1 ^2 * j3 ^2 + i1 ^2 * j4 ^2 + i2 ^2 * j1 ^2 + i2 ^2 * j2 ^2 + i2 ^2 * j3 ^2 + i2 ^2 * j4 ^2 + (i3 ^2 * j1 ^2 + i3 ^2 * j2 ^2 + i3 ^2 * (j3 ^2 + j4 ^2)) + (i4 ^2 * j1 ^2 + i4 ^2 * j2 ^2 + i4 ^2 * j3 ^2 + i4 ^2 * j4 ^2) + 2 * (i3 * j2 + i1 * j4) * (i2 * j3 + i4 * j1) + 2 * (i4 * j3 + i1 * j2) * (i2 * j1 + i3 * j4) + 2 * (i1 * j3 + i2 * j4) * (i3 * j1 + i4 * j2)

Rule argument

spec: int-mult name: dist-rsplit seq: ⊦ i * (j + j0) = i * j + i * j0 subst: [j0, j, i] → [j3 ^2 + j4 ^2, j2 ^2, i3 ^2] flags: <all paths>

Additional information

Used terms: j3 ^2 + j4 ^2, j2 ^2, i3 ^2