Let x0 of type ι be given.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
Apply nat_p_ordinal with
x0.
The subproof is completed by applying L1.
Apply ordinal_SNo with
x0.
The subproof is completed by applying L2.
Apply nat_ind with
λ x1 . add_nat x0 x1 = add_SNo x0 x1 leaving 2 subgoals.
Apply add_SNo_0R with
x0,
λ x1 x2 . add_nat x0 0 = x2 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying add_nat_0R with x0.
Let x1 of type ι be given.
Apply add_SNo_ordinal_SR with
x0,
x1,
λ x2 x3 . add_nat x0 (ordsucc x1) = x3 leaving 3 subgoals.
The subproof is completed by applying L2.
Apply nat_p_ordinal with
x1.
The subproof is completed by applying H4.
Apply H5 with
λ x2 x3 . add_nat x0 (ordsucc x1) = ordsucc x2.
Apply add_nat_SR with
x0,
x1.
The subproof is completed by applying H4.
Let x1 of type ι be given.
Apply L4 with
x1.
Apply omega_nat_p with
x1.
The subproof is completed by applying H5.