Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: x1 ∈ int.

Assume H1: gcd_reln x0 (add_SNo x1 (minus_SNo x0)) x2.

Apply H1 with gcd_reln x0 x1 x2.

Apply H2 with (∀ x3 . divides_int x3 x0 ⟶ divides_int x3 (add_SNo x1 (minus_SNo x0)) ⟶ SNoLe x3 x2) ⟶ gcd_reln x0 x1 x2.

Assume H3: divides_int x2 x0.

Assume H5: ∀ x3 . divides_int x3 x0 ⟶ divides_int x3 (add_SNo x1 (minus_SNo x0)) ⟶ SNoLe x3 x2.

Apply H3 with gcd_reln x0 x1 x2.

Assume H6: and (x2 ∈ int) (x0 ∈ int).

Assume H7: ∃ x3 . and (x3 ∈ int) (mul_SNo x2 x3 = x0).

Apply H6 with gcd_reln x0 x1 x2.

Assume H8: x2 ∈ int.

Assume H9: x0 ∈ int.

Apply and3I with divides_int x2 x0, divides_int x2 x1, ∀ x3 . divides_int x3 x0 ⟶ divides_int x3 x1 ⟶ SNoLe x3 x2 leaving 3 subgoals.

The subproof is completed by applying H3.

Apply add_SNo_minus_R2' with x1, x0, λ x3 x4 . divides_int x2 x3 leaving 3 subgoals.

Apply int_SNo with x1.

The subproof is completed by applying H0.

Apply int_SNo with x0.

The subproof is completed by applying H9.

Apply divides_int_add_SNo with x2, add_SNo x1 (minus_SNo x0), x0 leaving 2 subgoals.

The subproof is completed by applying H4.

The subproof is completed by applying H3.

Let x3 of type ι be given.

Assume H10: divides_int x3 x0.

Assume H11: divides_int x3 x1.

Apply H5 with x3 leaving 2 subgoals.

The subproof is completed by applying H10.

Apply divides_int_add_SNo with x3, x1, minus_SNo x0 leaving 2 subgoals.

The subproof is completed by applying H11.

Apply divides_int_minus_SNo with x3, x0.

The subproof is completed by applying H10.

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Proofgold Proof