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Proof step 71 from proof euler-four-square-identity (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 72:
⊦ (i + j)^2 = i ^2 + 2 * i * j + j ^2
Proof step 73:
⊦ 2 * (i1 * j2 + i2 * j1) * i3 * j4 + 2 * (i1 * j2) * i2 * j1 + ( ( (i4 * j1)^2 + 2 * (i4 * j1) * (i1 * j4 + i2 * j3) + (i1 * j4)^2 + 2 * (i1 * j4) * i2 * j3 + (i2 * j3)^2 - 2 * (i4 * j1 + i1 * j4 + i2 * j3) * i3 * j2) + (i3 * j2)^2) + 2 * (i2 * j2 + i3 * j3) * i4 * j4 + 2 * (i2 * j2) * i3 * j3 + (i4 * j2 + i3 * j1 + i1 * j3)^2 = (i3 ^2 + i4 ^2) * j1 ^2 + (j3 ^2 + j4 ^2) * i1 ^2 + (i3 ^2 + i4 ^2) * j2 ^2 + i2 ^2 * j3 ^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

Rule argument

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

Additional information

Used terms: i2 * j3, i1 * j4