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Proof step 17 from proof euler-four-square-identity (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ (i + j)^2 = i ^2 + 2 * i * j + j ^2
Proof step 19:
⊦ (i4 * j1 + i1 * j4 + i2 * j3 - i3 * j2)^2 + (i4 * j2 + i3 * j1 + i1 * j3 - i2 * j4)^2 + ( ( ((i1 * j2)^2 + 2 * (i1 * j2) * i2 * j1 + (i2 * j1)^2) + 2 * (i1 * j2 + i2 * j1) * i3 * j4 + (i3 * j4)^2 - 2 * ((i1 * j2 + i2 * j1) + i3 * j4) * i4 * j3) + (i4 * j3)^2) + ((i2 * j2 + i3 * j3) + i4 * j4)^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) + 2 * (i1 * j1) * ((i2 * j2 + i3 * j3) + i4 * j4)

Rule argument

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

Additional information

Used terms: i2 * j1, i1 * j2