Apply add_nat_SR with 11, 0, λ x0 x1 . x1 = 12 leaving 2 subgoals.

The subproof is completed by applying nat_0.

set y0 to be ordsucc (add_nat 11 0)

set y1 to be 12

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

Let x2 of type ι → ο be given.

Assume H0: x2 12.

set y3 to be λ x3 . x2

Apply add_nat_0R with 11, 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