Let x0 of type ι be given.
Let x1 of type ι be given.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Apply ordinal_SNo with
x1.
The subproof is completed by applying H1.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Apply ordinal_ordsucc with
x1.
The subproof is completed by applying H1.
Apply ordinal_SNo with
ordsucc x1.
The subproof is completed by applying L5.
Apply add_SNo_com with
x0,
ordsucc x1,
λ x2 x3 . x3 = ordsucc (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying L4.
The subproof is completed by applying L6.
Apply add_SNo_ordinal_SL with
x1,
x0,
λ x2 x3 . x3 = ordsucc (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply add_SNo_com with
x1,
x0,
λ x2 x3 . ordsucc x3 = ordsucc (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L4.
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying H7.