Let x0 of type ι be given.
Let x1 of type ι be given.
Claim L2:
∀ x3 : ι → ο . x3 y2 ⟶ x3 (add_nat x0 x1)Let x3 of type ι → ο be given.
Apply add_nat_add_SNo with
x1,
y2,
λ x4 . x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_add_CSNo with
x1,
y2,
λ x4 . x3 leaving 3 subgoals.
Apply omega_SNo with
x1.
The subproof is completed by applying H0.
Apply omega_SNo with
y2.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x3 of type ι → ι → ο be given.
Apply L2 with
λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H3: x3 y2 y2.
The subproof is completed by applying H3.