Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNoLeE with
x2,
add_SNo x0 (minus_SNo x1),
SNoLe (add_SNo x2 x1) x0 leaving 5 subgoals.
The subproof is completed by applying H2.
Apply SNo_add_SNo with
x0,
minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply SNoLtLe with
add_SNo x2 x1,
x0.
Apply add_SNo_minus_Lt2 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 (add_SNo x4 x1) x0.
Apply add_SNo_minus_R2' with
x0,
x1,
λ x3 x4 . SNoLe x4 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNoLe_ref with x0.