Let x0 of type ι be given.
Apply xm with
SNoLe 0 x0,
SNoLev (abs_SNo x0) = SNoLev x0 leaving 2 subgoals.
Apply nonneg_abs_SNo with
x0,
λ x1 x2 . SNoLev x2 = SNoLev x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι → ι → ο be given.
The subproof is completed by applying H2.
Apply not_nonneg_abs_SNo with
x0,
λ x1 x2 . SNoLev x2 = SNoLev x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply minus_SNo_Lev with
x0.
The subproof is completed by applying H0.