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

Let x0 of type ι be given.

Assume H0: x0 ∈ 0.

Apply FalseE with ∃ x1 . and (x1 ∈ omega) (add_nat x0 x1 = 0).

Apply EmptyE with x0.

The subproof is completed by applying H0.

Let x0 of type ι be given.

Assume H0: nat_p x0.

Assume H1: ∀ x1 . x1 ∈ x0 ⟶ ∃ x2 . and (x2 ∈ omega) (add_nat x1 x2 = x0).

Let x1 of type ι be given.

Assume H2: x1 ∈ ordsucc x0.

Apply ordsuccE with x0, x1, ∃ x2 . and (x2 ∈ omega) (add_nat x1 x2 = ordsucc x0) leaving 3 subgoals.

The subproof is completed by applying H2.

Assume H3: x1 ∈ x0.

Apply H1 with x1, ∃ x2 . and (x2 ∈ omega) (add_nat x1 x2 = ordsucc x0) leaving 2 subgoals.

The subproof is completed by applying H3.

Let x2 of type ι be given.

Assume H4: (λ x3 . and (x3 ∈ omega) (add_nat x1 x3 = x0)) x2.

Apply H4 with ∃ x3 . and (x3 ∈ omega) (add_nat x1 x3 = ordsucc x0).

Assume H5: x2 ∈ omega.

Assume H6: add_nat x1 x2 = x0.

Let x3 of type ο be given.

Assume H7: ∀ x4 . and (x4 ∈ omega) (add_nat x1 x4 = ordsucc x0) ⟶ x3.

Apply H7 with ordsucc x2.

Apply andI with ordsucc x2 ∈ omega, add_nat x1 (ordsucc x2) = ordsucc x0 leaving 2 subgoals.

Apply omega_ordsucc with x2.

The subproof is completed by applying H5.

Apply add_nat_SR with x1, x2, λ x4 x5 . x5 = ordsucc x0 leaving 2 subgoals.

Apply omega_nat_p with x2.

The subproof is completed by applying H5.

set y4 to be ordsucc (add_nat x1 x2)

set y5 to be ordsucc x1

Claim L8: ∀ x6 : ι → ο . x6 y5 ⟶ x6 y4

Let x6 of type ι → ο be given.

Assume H8: x6 (ordsucc x2).

set y7 to be λ x7 . x6

Apply H6 with λ x8 x9 . y7 (ordsucc x8) (ordsucc x9).

The subproof is completed by applying H8.

Let x6 of type ι → ι → ο be given.

Apply L8 with λ x7 . x6 x7 y5 ⟶ x6 y5 x7.

Assume H9: x6 y5 y5.

The subproof is completed by applying H9.

Assume H3: x1 = x0.

Let x2 of type ο be given.

Assume H4: ∀ x3 . and (x3 ∈ omega) (add_nat x1 x3 = ordsucc x0) ⟶ x2.

Apply H4 with 1.

Apply andI with 1 ∈ omega, add_nat x1 1 = ordsucc x0 leaving 2 subgoals.

Apply nat_p_omega with 1.

The subproof is completed by applying nat_1.

Apply H3 with λ x3 x4 . add_nat x1 1 = ordsucc x3.

Apply add_nat_SR with x1, 0, λ x3 x4 . x4 = ordsucc x1 leaving 2 subgoals.

The subproof is completed by applying nat_0.

set y3 to be ...

set y4 to be ordsucc ...

Claim L5: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3

Let x5 of type ι → ο be given.

Assume H5: x5 (ordsucc y3).

set y6 to be λ x6 . x5

Apply add_nat_0R with y3, λ x7 x8 . y6 (ordsucc x7) (ordsucc x8).

The subproof is completed by applying H5.

Let x5 of type ι → ι → ο be given.

Apply L5 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.

Assume H6: x5 y4 y4.

The subproof is completed by applying H6.

■

Proofgold Proof