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Proof step 10 from proof euler-four-square-identity (previous step)

⊦ (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)^2 - 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4)) + ((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 11:
⊦ (i * j)^2 = i ^2 * j ^2
Proof step 12:
⊦ (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 ^2 * j1 ^2 - 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4)) + ((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: <toplevel> name: two-mult seq: ⊦ (i * j)^2 = i ^2 * j ^2 subst: [j, i] → [j1, i1] flags: <all paths>

Additional information

Used terms: j1, i1