Apply nat_complete_ind with λ x0 . ∀ x1 . equip x0 x1 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 (x2 x3) = x3) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 x3 = x3 ⟶ ∀ x4 : ο . x4) ⟶ even_nat x0.

Let x0 of type ι be given.

Assume H0: nat_p x0.

Assume H1: ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . equip x1 x2 ⟶ ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ (∀ x4 . x4 ∈ x2 ⟶ x3 (x3 x4) = x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ x3 x4 = x4 ⟶ ∀ x5 : ο . x5) ⟶ even_nat x1.

Let x1 of type ι be given.

Assume H2: equip x0 x1.

Let x2 of type ι → ι be given.

Assume H3: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1.

Assume H4: ∀ x3 . x3 ∈ x1 ⟶ x2 (x2 x3) = x3.

Assume H5: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 = x3 ⟶ ∀ x4 : ο . x4.

Apply H2 with even_nat x0.

Let x3 of type ι → ι be given.

Assume H6: bij x0 x1 x3.

Apply H6 with even_nat x0.

Assume H7: and (∀ x4 . x4 ∈ x0 ⟶ x3 x4 ∈ x1) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5).

Apply H7 with (∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x3 x5 = x4)) ⟶ even_nat x0.

Assume H8: ∀ x4 . x4 ∈ x0 ⟶ x3 x4 ∈ x1.

Assume H9: ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5.

Assume H10: ∀ x4 . x4 ∈ x1 ⟶ ∃ x5 . and (x5 ∈ x0) (x3 x5 = x4).

Apply nat_inv with x0, even_nat x0 leaving 3 subgoals.

The subproof is completed by applying H0.

Assume H11: x0 = 0.

Apply H11 with λ x4 x5 . even_nat x5.

The subproof is completed by applying even_nat_0.

Assume H11: ∃ x4 . and (nat_p x4) (x0 = ordsucc x4).

Apply H11 with even_nat x0.

Let x4 of type ι be given.

Assume H12: (λ x5 . and (nat_p x5) (x0 = ordsucc x5)) x4.

Apply H12 with even_nat x0.

Assume H13: nat_p x4.

Assume H14: x0 = ordsucc x4.

Claim L15: ...

...

Apply H10 with x2 (x3 x4), even_nat x0 leaving 2 subgoals.

Apply H3 with x3 x4.

Apply H8 with x4.

The subproof is completed by applying L15.

Let x5 of type ι be given.

Assume H16: (λ x6 . and (x6 ∈ x0) (x3 x6 = x2 (x3 x4))) x5.

Apply H16 with even_nat x0.

Assume H17: x5 ∈ x0.

Assume H18: x3 x5 = x2 (x3 x4).

Apply ordsuccE with x4, x5, even_nat ... leaving 3 subgoals.

...

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Proofgold Proof