Let x0 of type ι → ο be given.
Assume H0: x0 0.
Apply andI with
nat_p 0,
x0 0 leaving 2 subgoals.
The subproof is completed by applying nat_0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Apply H3 with
and (nat_p (ordsucc x1)) (x0 (ordsucc x1)).
Assume H5: x0 x1.
Apply andI with
nat_p (ordsucc x1),
x0 (ordsucc x1) leaving 2 subgoals.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H4.
Apply H1 with
x1 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Let x1 of type ι be given.
Apply H4 with
λ x2 . and (nat_p x2) (x0 x2) leaving 2 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L3.
Apply andER with
nat_p x1,
x0 x1.
The subproof is completed by applying L5.