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Proof step 2 from proof euler-four-square-identity2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ (i - j)^2 = (i ^2 - 2 * i * j) + j ^2
Proof step 4:
⊦ (i1 ^2 + i2 ^2 + i3 ^2 + i4 ^2) * (j1 ^2 + j2 ^2 + j3 ^2 + j4 ^2) = (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + (i4 * j3 + i1 * j2 - i2 * j1 + i3 * j4)^2 + (i1 * j3 + i2 * j4 - i3 * j1 + i4 * j2)^2 + ((i3 * j2 + i1 * j4)^2 - 2 * (i3 * j2 + i1 * j4) * (i2 * j3 + i4 * j1)) + (i2 * j3 + i4 * j1)^2

Rule argument

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

Additional information

Used terms: i2 * j3 + i4 * j1, i3 * j2 + i1 * j4