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Proof step 92 from proof euler-four-square-identity (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 93:
⊦ (i * j)^2 = i ^2 * j ^2
Proof step 94:
⊦ (i4 ^2 * j2 ^2 + 2 * (i4 * j2) * (i3 * j1 + i1 * j3) + (i3 * j1 + i1 * j3)^2) + 2 * (i2 * j2) * i3 * j3 + 2 * (i2 * j2 + i3 * j3) * i4 * j4 + 2 * (i1 * j2) * i2 * j1 + 2 * (i1 * j2 + i2 * j1) * i3 * j4 + 2 * (i4 * j1) * (i1 * j4 + i2 * j3) + 2 * (i1 * j4) * i2 * j3 = i3 ^2 * j1 ^2 + j3 ^2 * i1 ^2 + i4 ^2 * j2 ^2 + 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

Rule argument

spec: <toplevel> name: two-mult seq: ⊦ (i * j)^2 = i ^2 * j ^2 subst: [j, i] → [j2, i4] flags: <all paths>

Additional information

Used terms: j2, i4