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_nat x0 (add_SNo x1 x2).
Apply H3 with
(∃ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1)) ⟶ divides_nat x0 (add_SNo x1 x2).
Apply H6 with
divides_nat x0 (add_SNo x1 x2).
Let x3 of type ι be given.
Apply H7 with
divides_nat x0 (add_SNo x1 x2).
Apply H2 with
divides_nat x0 (add_SNo x1 x2).
Apply H10 with
(∃ x4 . and (x4 ∈ omega) (mul_nat x0 x4 = x2)) ⟶ divides_nat x0 (add_SNo x1 x2).
Apply H13 with
divides_nat x0 (add_SNo x1 x2).
Let x4 of type ι be given.
Apply H14 with
divides_nat x0 (add_SNo x1 x2).
Apply add_nat_add_SNo with
x1,
x2,
λ x5 x6 . and (and (x0 ∈ omega) (x5 ∈ omega)) (∃ x7 . and (x7 ∈ omega) (mul_nat x0 x7 = x5)) leaving 3 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H12.
Apply and3I with
x0 ∈ omega,
add_nat x1 x2 ∈ omega,
∃ x5 . and (x5 ∈ omega) (mul_nat x0 x5 = add_nat x1 x2) leaving 3 subgoals.
The subproof is completed by applying H4.
Apply nat_p_omega with
add_nat x1 x2.
Apply add_nat_p with
x1,
x2 leaving 2 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H5.
Apply omega_nat_p with
x2.
The subproof is completed by applying H12.
Let x5 of type ο be given.
Apply H17 with
add_nat x3 x4.
Apply andI with
add_nat x3 x4 ∈ omega,
mul_nat x0 (add_nat x3 x4) = add_nat x1 x2 leaving 2 subgoals.
Apply nat_p_omega with
add_nat x3 x4.
Apply add_nat_p with
x3,
x4 leaving 2 subgoals.
Apply omega_nat_p with
x3.
The subproof is completed by applying H8.
Apply omega_nat_p with
x4.
The subproof is completed by applying H15.
Apply mul_add_nat_distrL with
x0,
x3,
x4,
λ x6 x7 . x7 = add_nat x1 x2 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply omega_nat_p with
x3.
The subproof is completed by applying H8.
Apply omega_nat_p with
x4.
The subproof is completed by applying H15.
set y6 to be ...
set y7 to be ...
Let x8 of type ι → ι → ο be given.
Apply L18 with
λ x9 . x8 x9 y7 ⟶ x8 y7 x9.
Assume H19: x8 ... ....