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Proof step 10 from proof lagrange-primedivisors (previous step)

Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, ∀ j0. j0 ∈ ints0 → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i0, i1, i2, i3. (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 11:
⊦ (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 + ((i1 * j2 - i2 * j1 - i3 * j4) + i4 * j3)^2 + (i1 * j3 + i2 * j4 - i3 * j1 - i4 * j2)^2 + ((i1 * j4 - i2 * j3) + i3 * j2 - i4 * j1)^2
Proof step 12:
(i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * (i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2) = (((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3)^2 + ((i4 * i1 - i5 * i0 - i6 * i3) + i7 * i2)^2 + (i4 * i2 + i5 * i3 - i6 * i0 - i7 * i1)^2 + ((i4 * i3 - i5 * i2) + i6 * i1 - i7 * i0)^2, Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, ∀ j0. j0 ∈ ints0 → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i0, i1, i2, i3. (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2

Rule argument

spec: int-square inst: name: euler-four-square-identity2 seq: ⊦ (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 + ((i1 * j2 - i2 * j1 - i3 * j4) + i4 * j3)^2 + (i1 * j3 + i2 * j4 - i3 * j1 - i4 * j2)^2 + ((i1 * j4 - i2 * j3) + i3 * j2 - i4 * j1)^2 subst: [i1, i2, i3, i4, j1, j2, j3, j4] → [i4, i5, i6, i7, i0, i1, i2, i3] flags: []

Additional information

Used terms: i4, i5, i6, i7, i0, i1, i2, i3