Apply add_nat_SR with 11, 1, λ x0 x1 . x1 = 13 leaving 2 subgoals.

The subproof is completed by applying nat_1.

set y0 to be ordsucc (add_nat 11 1)

set y1 to be 13

Claim L0: ∀ x2 : ι → ο . x2 y1 ⟶ x2 y0

Let x2 of type ι → ο be given.

Assume H0: x2 13.

set y3 to be λ x3 . x2

Apply unknownprop_e8b8cc3197e9023162add07151dc0f7ca1683c2ac8baa030b8f4fa51ef79c8ab with λ x4 x5 . y3 (ordsucc x4) (ordsucc x5).

The subproof is completed by applying H0.

Let x2 of type ι → ι → ο be given.

Apply L0 with λ x3 . x2 x3 y1 ⟶ x2 y1 x3.

Assume H1: x2 y1 y1.

The subproof is completed by applying H1.

■

Proofgold Proof