Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNoLeE with
x0,
add_SNo x2 x1,
SNoLe (add_SNo x0 (minus_SNo x1)) x2 leaving 5 subgoals.
The subproof is completed by applying H0.
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 H3.
Apply SNoLtLe with
add_SNo x0 (minus_SNo x1),
x2.
Apply add_SNo_minus_Lt1b 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 (minus_SNo x1)) x2.
Apply add_SNo_minus_R2 with
x2,
x1,
λ x3 x4 . SNoLe x4 x2 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.