Let x0 of type ι be given.
Let x1 of type ι be given.
Apply real_complete1 with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Apply add_nat_add_SNo with
x2,
1,
λ x3 x4 . ordsucc x2 = x3,
λ x3 x4 . and (and (SNoLe (ap x0 x2) (ap x1 x2)) (SNoLe (ap x0 x2) (ap x0 x4))) (SNoLe (ap x1 x4) (ap x1 x2)) leaving 4 subgoals.
The subproof is completed by applying H3.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Apply add_nat_SR with
x2,
0,
λ x3 x4 . ordsucc x2 = x4 leaving 2 subgoals.
The subproof is completed by applying nat_0.
Apply add_nat_0R with
x2,
λ x3 x4 . ordsucc x2 = ordsucc x4.
Let x4 of type ι → ι → ο be given.
Assume H4: x4 y3 y3.
The subproof is completed by applying H4.
Apply H2 with
x2.
The subproof is completed by applying H3.