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Proof step 87 from proof euler-four-square-identity2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 88:
⊦ (i * i0 + j * j0)^2 = i ^2 * i0 ^2 + 2 * i * i0 * j * j0 + j ^2 * j0 ^2
Proof step 89:
⊦ ((i1 ^2 * j1 ^2 + 2 * i1 * j1 * i2 * j2 + i2 ^2 * j2 ^2) + 2 * (i1 * j1 + i2 * j2) * (i3 * j3 + i4 * j4) + (i3 * j3 + i4 * j4)^2) + 2 * i1 * j3 * i2 * j4 + 2 * i3 * j2 * i1 * j4 + 2 * i3 * j1 * i4 * j2 + 2 * i2 * j1 * i3 * j4 + 2 * i2 * j3 * i4 * j1 + 2 * i4 * j3 * i1 * j2 = 2 * (i1 * j3 + i2 * j4) * (i3 * j1 + i4 * j2) + 2 * (i4 * j3 + i1 * j2) * (i2 * j1 + i3 * j4) + 2 * (i3 * j2 + i1 * j4) * (i2 * j3 + i4 * j1) + i4 ^2 * j4 ^2 + i3 ^2 * j3 ^2 + i2 ^2 * j2 ^2 + i1 ^2 * j1 ^2

Rule argument

spec: <toplevel> name: two-pm seq: ⊦ (i * i0 + j * j0)^2 = i ^2 * i0 ^2 + 2 * i * i0 * j * j0 + j ^2 * j0 ^2 subst: [j0, j, i0, i] → [j2, i2, j1, i1] flags: <all paths>

Additional information

Used terms: j2, i2, j1, i1