Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply nonneg_abs_SNo with
add_SNo x1 (minus_SNo x0),
λ x3 x4 . SNoLt x4 x2 leaving 2 subgoals.
Apply add_SNo_minus_SNo_rinv with
x0,
λ x3 x4 . SNoLe x3 (add_SNo x1 (minus_SNo x0)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Le1 with
x0,
minus_SNo x0,
x1 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply add_SNo_minus_Lt1b with
x1,
x0,
x2 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply add_SNo_com with
x2,
x0,
λ x3 x4 . SNoLt x1 x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
The subproof is completed by applying H4.