Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: x1 = 0 ⟶ ∀ x2 : ο . x2.

Assume H1: divides_nat x0 x1.

Apply H1 with x0 ⊆ x1.

Assume H2: and (x0 ∈ omega) (x1 ∈ omega).

Apply H2 with (∃ x2 . and (x2 ∈ omega) (mul_nat x0 x2 = x1)) ⟶ x0 ⊆ x1.

Assume H3: x0 ∈ omega.

Assume H4: x1 ∈ omega.

Assume H5: ∃ x2 . and (x2 ∈ omega) (mul_nat x0 x2 = x1).

Apply H5 with x0 ⊆ x1.

Let x2 of type ι be given.

Assume H6: (λ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1)) x2.

Apply H6 with x0 ⊆ x1.

Assume H7: x2 ∈ omega.

Assume H8: mul_nat x0 x2 = x1.

Apply H8 with λ x3 x4 . x0 ⊆ x3.

Apply unknownprop_0dc8d11d1ba28645d1565e6f95fe26f514da291413e114d0327c09556f7d23e9 with x0, x2 leaving 2 subgoals.

Apply omega_nat_p with x0.

The subproof is completed by applying H3.

Apply setminusI with omega, 1, x2 leaving 2 subgoals.

The subproof is completed by applying H7.

Assume H9: x2 ∈ 1.

Apply H0.

Apply H8 with λ x3 x4 . x3 = 0.

Apply cases_1 with x2, λ x3 . mul_nat x0 x3 = 0 leaving 2 subgoals.

The subproof is completed by applying H9.

The subproof is completed by applying mul_nat_0R with x0.

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Proofgold Proof