Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply add_SNo_Lt1_cancel with
add_SNo x2 x1,
minus_SNo x1,
x0 leaving 4 subgoals.
Apply SNo_add_SNo with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply add_SNo_assoc with
x2,
x1,
minus_SNo x1,
λ x3 x4 . SNoLt x3 (add_SNo x0 (minus_SNo x1)) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with
x1.
The subproof is completed by applying H1.
Apply add_SNo_minus_SNo_rinv with
x1,
λ x3 x4 . SNoLt (add_SNo x2 x4) (add_SNo x0 (minus_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying H1.
Apply add_SNo_0R with
x2,
λ x3 x4 . SNoLt x4 (add_SNo x0 (minus_SNo x1)) leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.