Let x0 of type ι be given.
Let x1 of type ι be given.
Apply add_nat_add_SNo with
x0,
x1,
λ x2 x3 . x2 ∈ omega leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply nat_p_omega with
add_nat x0 x1.
Apply add_nat_p with
x0,
x1 leaving 2 subgoals.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
Apply omega_nat_p with
x1.
The subproof is completed by applying H1.