Apply add_nat_SR with 23, 7, λ x0 x1 . x1 = 31 leaving 2 subgoals.

The subproof is completed by applying nat_7.

set y0 to be ordsucc (add_nat 23 7)

set y1 to be 31

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

Let x2 of type ι → ο be given.

Assume H0: x2 31.

set y3 to be λ x3 . x2

Apply unknownprop_0ee4fa6dcc95212d2cb99e26434cded17dc43098a534e31c835c593428bcb471 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