Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: divides_int x0 x1.

Assume H1: divides_int x0 x2.

Apply H0 with divides_int x0 (add_SNo x1 x2).

Assume H2: and (x0 ∈ int) (x1 ∈ int).

Apply H2 with (∃ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = x1)) ⟶ divides_int x0 (add_SNo x1 x2).

Assume H3: x0 ∈ int.

Assume H4: x1 ∈ int.

Assume H5: ∃ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = x1).

Apply H5 with divides_int x0 (add_SNo x1 x2).

Let x3 of type ι be given.

Assume H6: (λ x4 . and (x4 ∈ int) (mul_SNo x0 x4 = x1)) x3.

Apply H6 with divides_int x0 (add_SNo x1 x2).

Assume H7: x3 ∈ int.

Assume H8: mul_SNo x0 x3 = x1.

Apply H1 with divides_int x0 (add_SNo x1 x2).

Assume H9: and (x0 ∈ int) (x2 ∈ int).

Apply H9 with (∃ x4 . and (x4 ∈ int) (mul_SNo x0 x4 = x2)) ⟶ divides_int x0 (add_SNo x1 x2).

Assume H10: x0 ∈ int.

Assume H11: x2 ∈ int.

Assume H12: ∃ x4 . and (x4 ∈ int) (mul_SNo x0 x4 = x2).

Apply H12 with divides_int x0 (add_SNo x1 x2).

Let x4 of type ι be given.

Assume H13: (λ x5 . and (x5 ∈ int) (mul_SNo x0 x5 = x2)) x4.

Apply H13 with divides_int x0 (add_SNo x1 x2).

Assume H14: x4 ∈ int.

Assume H15: mul_SNo x0 x4 = x2.

Apply and3I with x0 ∈ int, add_SNo x1 x2 ∈ int, ∃ x5 . and (x5 ∈ int) (mul_SNo x0 x5 = add_SNo x1 x2) leaving 3 subgoals.

The subproof is completed by applying H3.

Apply int_add_SNo with x1, x2 leaving 2 subgoals.

The subproof is completed by applying H4.

The subproof is completed by applying H11.

Let x5 of type ο be given.

Assume H16: ∀ x6 . and (x6 ∈ int) (mul_SNo x0 x6 = add_SNo x1 x2) ⟶ x5.

Apply H16 with add_SNo x3 x4.

Apply andI with add_SNo x3 x4 ∈ int, mul_SNo x0 (add_SNo x3 x4) = add_SNo x1 x2 leaving 2 subgoals.

Apply int_add_SNo with x3, x4 leaving 2 subgoals.

The subproof is completed by applying H7.

The subproof is completed by applying H14.

Apply mul_SNo_distrL with x0, x3, x4, λ x6 x7 . x7 = add_SNo x1 x2 leaving 4 subgoals.

Apply int_SNo with x0.

The subproof is completed by applying H3.

Apply int_SNo with x3.

The subproof is completed by applying H7.

Apply int_SNo with x4.

The subproof is completed by applying H14.

set y6 to be ...

set y7 to be ...

Claim L17: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6

Let x8 of type ι → ο be given.

Assume H17: x8 (add_SNo x3 x4).

set y9 to be ...

Apply H8 with λ x10 x11 . y9 (add_SNo x10 (mul_SNo x2 y6)) (add_SNo x11 (mul_SNo x2 y6)).

set y10 to be ...

Apply H15 with λ x11 x12 . y10 (add_SNo x4 ...) ....

...

Let x8 of type ι → ι → ο be given.

Apply L17 with λ x9 . x8 x9 y7 ⟶ x8 y7 x9.

Assume H18: x8 y7 y7.

The subproof is completed by applying H18.

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Proofgold Proof