Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply add_SNo_minus_R2 with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply add_SNo_minus_R2 with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply L4 with
λ x3 x4 . SNoLt x3 x2.
Apply L5 with
λ x3 x4 . SNoLt (add_SNo (add_SNo x0 x1) (minus_SNo x1)) x3.
Apply add_SNo_Lt1 with
add_SNo x0 x1,
minus_SNo x1,
add_SNo x2 x1 leaving 4 subgoals.
Apply SNo_add_SNo with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
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.