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Proof step 7 from proof Lagrange-SumOf4Squares (previous step)

Π all_prime_divisors(j) = j, 0 < j ⊦ j = 0, ∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2

User applied rule insert lemma

And obtained the following new goals

Proof step 8:
∀ j. j ∈ ints → (∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i, i0, i1, i2. Π ints = i ^2 + i0 ^2 + i1 ^2 + i2 ^2
Proof step 9:
Π all_prime_divisors(j) = j, 0 < j ⊦ ∀ j0. j0 ∈ all_prime_divisors(j) → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2), j = 0, ∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2
Proof step 15:
∃ i, i0, i1, i2. Π all_prime_divisors(j) = i ^2 + i0 ^2 + i1 ^2 + i2 ^2, Π all_prime_divisors(j) = j, 0 < j ⊦ j = 0, ∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2

Rule argument

spec: inst: name: lagrange-primedivisors seq: ∀ j. j ∈ ints → (∃ i, i0, i1, i2. j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i, i0, i1, i2. Π ints = i ^2 + i0 ^2 + i1 ^2 + i2 ^2 subst: [ints] → [all_prime_divisors(j)] flags: current, []

Additional information

Used terms: all_prime_divisors(j)