Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNoLeE with
add_SNo x2 x1,
x0,
SNoLe x2 (add_SNo x0 (minus_SNo x1)) leaving 5 subgoals.
Apply SNo_add_SNo with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply SNoLtLe with
x2,
add_SNo x0 (minus_SNo x1).
Apply add_SNo_minus_Lt2b with
x0,
x1,
x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply H4 with
λ x3 x4 . SNoLe x2 (add_SNo x3 (minus_SNo x1)).
Apply add_SNo_minus_R2 with
x2,
x1,
λ x3 x4 . SNoLe x2 x4 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying SNoLe_ref with x2.