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 x0 (minus_SNo x1),
x2,
SNoLe x0 (add_SNo x2 x1) leaving 5 subgoals.
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 H2.
The subproof is completed by applying H3.
Apply SNoLtLe with
x0,
add_SNo x2 x1.
Apply add_SNo_minus_Lt1 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 x0 (add_SNo x3 x1).
Apply add_SNo_minus_R2' with
x0,
x1,
λ x3 x4 . SNoLe x0 x4 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.