Let x0 of type ι be given.

Assume H0: x0 ∈ setminus omega (Sing 0).

Apply setminusE with omega, Sing 0, x0, ∀ x1 . nat_p x1 ⟶ ∃ x2 . and (x2 ∈ omega) (∃ x3 . and (x3 ∈ x0) (x1 = add_nat (mul_nat x2 x0) x3)) leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: x0 ∈ omega.

Assume H2: nIn x0 (Sing 0).

Claim L3: ...

...

Claim L4: ...

...

Claim L5: ...

...

Apply nat_ind with λ x1 . ∃ x2 . and (x2 ∈ omega) (∃ x3 . and (x3 ∈ x0) (x1 = add_nat (mul_nat x2 x0) x3)) leaving 2 subgoals.

Let x1 of type ο be given.

Assume H6: ∀ x2 . and (x2 ∈ omega) (∃ x3 . and (x3 ∈ x0) (0 = add_nat (mul_nat x2 x0) x3)) ⟶ x1.

Apply H6 with 0.

Apply andI with 0 ∈ omega, ∃ x2 . and (x2 ∈ x0) (0 = add_nat (mul_nat 0 x0) x2) leaving 2 subgoals.

Apply nat_p_omega with 0.

The subproof is completed by applying nat_0.

Let x2 of type ο be given.

Assume H7: ∀ x3 . and (x3 ∈ x0) (0 = add_nat (mul_nat 0 x0) x3) ⟶ x2.

Apply H7 with 0.

Apply andI with 0 ∈ x0, 0 = add_nat (mul_nat 0 x0) 0 leaving 2 subgoals.

The subproof is completed by applying L5.

Apply mul_nat_0L with x0, λ x3 x4 . 0 = add_nat x4 0 leaving 2 subgoals.

The subproof is completed by applying L3.

Let x3 of type ι → ι → ο be given.

The subproof is completed by applying add_nat_0R with 0, λ x4 x5 . x3 x5 x4.

Let x1 of type ι be given.

Assume H6: nat_p x1.

Assume H7: ∃ x2 . and (x2 ∈ omega) (∃ x3 . and (x3 ∈ x0) (x1 = add_nat (mul_nat x2 x0) x3)).

Apply H7 with ∃ x2 . and (x2 ∈ omega) (∃ x3 . and (x3 ∈ x0) (ordsucc x1 = add_nat (mul_nat x2 x0) x3)).

Let x2 of type ι be given.

Assume H8: (λ x3 . and (x3 ∈ omega) (∃ x4 . and (x4 ∈ x0) (x1 = add_nat (mul_nat x3 x0) x4))) x2.

Apply H8 with ∃ x3 . and (x3 ∈ omega) (∃ x4 . and (x4 ∈ x0) (ordsucc x1 = add_nat (mul_nat x3 x0) x4)).

Assume H9: x2 ∈ omega.

Assume H10: ∃ x3 . and (x3 ∈ x0) (x1 = add_nat (mul_nat x2 x0) x3).

Apply H10 with ∃ x3 . and (x3 ∈ omega) (∃ x4 . and (x4 ∈ x0) (ordsucc x1 = add_nat (mul_nat x3 x0) x4)).

Let x3 of type ι be given.

Assume H11: (λ x4 . and (x4 ∈ x0) (x1 = add_nat (mul_nat x2 x0) x4)) x3.

Apply H11 with ∃ x4 . and (x4 ∈ omega) (∃ x5 . and (x5 ∈ x0) (ordsucc x1 = add_nat (mul_nat x4 x0) ...)).

...

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Proofgold Proof