Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply minus_SNo_invol with
add_SNo x2 (minus_SNo x1),
λ x3 x4 . SNoLe x3 (minus_SNo x0) leaving 2 subgoals.
Apply SNo_add_SNo with
x2,
minus_SNo x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply minus_add_SNo_distr with
x2,
minus_SNo x1,
λ x3 x4 . SNoLe (minus_SNo x4) (minus_SNo x0) leaving 3 subgoals.
The subproof is completed by applying H2.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply minus_SNo_invol with
x1,
λ x3 x4 . SNoLe (minus_SNo (add_SNo (minus_SNo x2) x4)) (minus_SNo x0) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply minus_SNo_Le_contra with
x0,
add_SNo (minus_SNo x2) x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_add_SNo with
minus_SNo x2,
x1 leaving 2 subgoals.
Apply SNo_minus_SNo with
x2.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply add_SNo_com with
minus_SNo x2,
x1,
λ x3 x4 . SNoLe x0 x4 leaving 3 subgoals.
Apply SNo_minus_SNo with
x2.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
The subproof is completed by applying H3.