Let x0 of type ι be given.
Apply setminusE with
omega,
Sing 0,
x0,
∀ x1 . x1 ∈ int ⟶ ∃ x2 . and (x2 ∈ int) (∃ x3 . and (x3 ∈ x0) (x1 = add_SNo (mul_SNo x2 x0) x3)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply int_SNo_cases with
λ x1 . ∃ x2 . and (x2 ∈ int) (∃ x3 . and (x3 ∈ x0) (x1 = add_SNo (mul_SNo x2 x0) x3)) leaving 2 subgoals.
Let x1 of type ι be given.
Apply quotient_remainder_nat with
x0,
x1,
∃ x2 . and (x2 ∈ int) (∃ x3 . and (x3 ∈ x0) (x1 = add_SNo (mul_SNo x2 x0) x3)) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply omega_nat_p with
x1.
The subproof is completed by applying H5.
Let x2 of type ι be given.
Apply H6 with
∃ x3 . and (x3 ∈ int) (∃ x4 . and (x4 ∈ x0) (x1 = add_SNo (mul_SNo x3 x0) x4)).
Apply H8 with
∃ x3 . and (x3 ∈ int) (∃ x4 . and (x4 ∈ x0) (x1 = add_SNo (mul_SNo x3 x0) x4)).
Let x3 of type ι be given.
Apply H9 with
∃ x4 . and (x4 ∈ int) (∃ x5 . and (x5 ∈ x0) (x1 = add_SNo (mul_SNo x4 x0) x5)).
Assume H10: x3 ∈ x0.
Let x4 of type ο be given.
Apply H12 with
x2.
Apply andI with
x2 ∈ int,
∃ x5 . and (x5 ∈ x0) (x1 = add_SNo (mul_SNo x2 x0) x5) leaving 2 subgoals.
Apply Subq_omega_int with
x2.
The subproof is completed by applying H7.
Let x5 of type ο be given.
Apply H13 with
x3.
Apply andI with
x3 ∈ x0,
x1 = add_SNo (mul_SNo x2 x0) x3 leaving 2 subgoals.
The subproof is completed by applying H10.
set y6 to be ...
Claim L14: ∀ x7 : ι → ο . x7 y6 ⟶ x7 x1
Let x7 of type ι → ι → ο be given.
Apply L14 with
λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H15: x7 y6 y6.
The subproof is completed by applying H15.