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Proof step 37 from proof euler-four-square-identity (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ (i * j)^2 = i ^2 * j ^2
Proof step 39:
⊦ ((i2 * j2 + i3 * j3) + i4 * j4)^2 + i4 ^2 * j3 ^2 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2 + 2 * (i1 * j2) * i2 * j1 + 2 * (i1 * j2 + i2 * j1) * i3 * j4 + (i3 * j4)^2 = (i3 ^2 + i4 ^2) * j1 ^2 + (j3 ^2 + j4 ^2) * i1 ^2 + ((i2 ^2 + i3 ^2 + i4 ^2) * j2 ^2 + (i2 ^2 * j3 ^2 + i3 ^2 * j3 ^2 + i4 ^2 * j3 ^2) + (i2 ^2 + i3 ^2 + i4 ^2) * j4 ^2) + 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4) + 2 * ((i1 * j2 + i2 * j1) + i3 * j4) * i4 * j3

Rule argument

spec: <toplevel> name: two-mult seq: ⊦ (i * j)^2 = i ^2 * j ^2 subst: [j, i] → [j3, i4] flags: <all paths>

Additional information

Used terms: j3, i4