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Proof step 13 from proof lagrange-primedivisors (previous step)

Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = (((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3)^2 + (i7 * i2 + i4 * i1 - i5 * i0 + i6 * i3)^2 + (i4 * i2 + i5 * i3 - i6 * i0 + i7 * i1)^2 + (i6 * i1 + i4 * i3 - i5 * i2 + i7 * i0)^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

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: i6 * i1 + i4 * i3 - i5 * i2 + i7 * i0, i4 * i2 + i5 * i3 - i6 * i0 + i7 * i1, i7 * i2 + i4 * i1 - i5 * i0 + i6 * i3, ((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3

Additional information

Used simplifier rules: add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i6 * i1 + i4 * i3 - i5 * i2 + i7 * i0, i4 * i2 + i5 * i3 - i6 * i0 + i7 * i1, i7 * i2 + i4 * i1 - i5 * i0 + i6 * i3, ((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3