Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply H1 with
divides_int x0 (mul_SNo x1 x2).
Apply H2 with
(∃ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = x1)) ⟶ divides_int x0 (mul_SNo x1 x2).
Apply H5 with
divides_int x0 (mul_SNo x1 x2).
Let x3 of type ι be given.
Apply H6 with
divides_int x0 (mul_SNo x1 x2).
Apply and3I with
x0 ∈ int,
mul_SNo x1 x2 ∈ int,
∃ x4 . and (x4 ∈ int) (mul_SNo x0 x4 = mul_SNo x1 x2) leaving 3 subgoals.
The subproof is completed by applying H3.
Apply int_mul_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H0.
Let x4 of type ο be given.
Apply H9 with
mul_SNo x3 x2.
Apply andI with
mul_SNo x3 x2 ∈ int,
mul_SNo x0 (mul_SNo x3 x2) = mul_SNo x1 x2 leaving 2 subgoals.
Apply int_mul_SNo with
x3,
x2 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H0.
Apply mul_SNo_assoc with
x0,
x3,
x2,
λ x5 x6 . x6 = mul_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
x2.
The subproof is completed by applying H0.
Claim L10: ∀ x7 : ι → ο . x7 y6 ⟶ x7 y5
Let x7 of type ι → ο be given.
set y8 to be λ x8 . x7
Apply H8 with
λ x9 x10 . y8 (mul_SNo x9 x4) (mul_SNo x10 x4).
The subproof is completed by applying H10.
Let x7 of type ι → ι → ο be given.
Apply L10 with
λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H11: x7 y6 y6.
The subproof is completed by applying H11.