Apply add_nat_SR with 23, 12, λ x0 x1 . x1 = 36 leaving 2 subgoals.

The subproof is completed by applying nat_12.

set y0 to be ordsucc (add_nat 23 12)

set y1 to be 36

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

Let x2 of type ι → ο be given.

Assume H0: x2 36.

set y3 to be λ x3 . x2

Apply unknownprop_564d072b65cfaa0a3fa2c4ea6b39aa7d06f0066a56a783eab00e49c33252e1e4 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