Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_p_ordinal with
x0.
The subproof is completed by applying H0.
Apply nat_p_SNo with
x0.
The subproof is completed by applying H0.
Apply nat_p_ordinal with
x1.
The subproof is completed by applying H1.
Apply add_SNo_com with
x0,
1,
λ x2 x3 . SNoLe x3 x1 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying SNo_1.
Apply ordinal_ordsucc_SNo_eq with
x0,
λ x2 x3 . SNoLe x2 x1 leaving 2 subgoals.
The subproof is completed by applying L2.
Apply ordinal_SNoLev_max_2 with
x1,
ordsucc x0 leaving 3 subgoals.
The subproof is completed by applying L4.
Apply nat_p_SNo with
ordsucc x0.
Apply nat_ordsucc with
x0.
The subproof is completed by applying H0.
Apply ordinal_SNoLev with
ordsucc x0,
λ x2 x3 . x3 ∈ ordsucc x1 leaving 2 subgoals.
Apply ordinal_ordsucc with
x0.
The subproof is completed by applying L2.
Apply ordinal_ordsucc_In with
x1,
x0 leaving 2 subgoals.
The subproof is completed by applying L4.
Apply ordinal_SNoLt_In with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L4.
The subproof is completed by applying H5.