Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: gcd_reln x0 x1 x2.

Apply H0 with gcd_reln x0 (minus_SNo x1) x2.

Assume H1: and (divides_int x2 x0) (divides_int x2 x1).

Apply H1 with (∀ x3 . divides_int x3 x0 ⟶ divides_int x3 x1 ⟶ SNoLe x3 x2) ⟶ gcd_reln x0 (minus_SNo x1) x2.

Assume H2: divides_int x2 x0.

Assume H3: divides_int x2 x1.

Assume H4: ∀ x3 . divides_int x3 x0 ⟶ divides_int x3 x1 ⟶ SNoLe x3 x2.

Apply H3 with and (and (divides_int x2 x0) (divides_int x2 (minus_SNo x1))) (∀ x3 . divides_int x3 x0 ⟶ divides_int x3 (minus_SNo x1) ⟶ SNoLe x3 x2).

Assume H5: and (x2 ∈ int) (x1 ∈ int).

Assume H6: ∃ x3 . and (x3 ∈ int) (mul_SNo x2 x3 = x1).

Apply H5 with and (and (divides_int x2 x0) (divides_int x2 (minus_SNo x1))) (∀ x3 . divides_int x3 x0 ⟶ divides_int x3 (minus_SNo x1) ⟶ SNoLe x3 x2).

Assume H7: x2 ∈ int.

Assume H8: x1 ∈ int.

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

The subproof is completed by applying H2.

Apply divides_int_minus_SNo with x2, x1.

The subproof is completed by applying H3.

Let x3 of type ι be given.

Assume H9: divides_int x3 x0.

Assume H10: divides_int x3 (minus_SNo x1).

Apply H4 with x3 leaving 2 subgoals.

The subproof is completed by applying H9.

Apply minus_SNo_invol with x1, λ x4 x5 . divides_int x3 x4 leaving 2 subgoals.

Apply int_SNo with x1.

The subproof is completed by applying H8.

Apply divides_int_minus_SNo with x3, minus_SNo x1.

The subproof is completed by applying H10.

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Proofgold Proof