Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply SNo_mul_SNo with
eps_ 1,
x0 leaving 2 subgoals.
The subproof is completed by applying SNo_eps_1.
The subproof is completed by applying H0.
Apply SNoLeLt_tra with
add_SNo x1 x2,
add_SNo (mul_SNo (eps_ 1) x0) x2,
x0 leaving 5 subgoals.
Apply SNo_add_SNo with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply SNo_add_SNo with
mul_SNo (eps_ 1) x0,
x2 leaving 2 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
Apply add_SNo_Le1 with
x1,
x2,
mul_SNo (eps_ 1) x0 leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L5.
The subproof is completed by applying H3.
Apply eps_1_split_eq with
x0,
λ x3 x4 . SNoLt (add_SNo (mul_SNo (eps_ 1) x0) x2) x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Lt2 with
mul_SNo (eps_ 1) x0,
x2,
mul_SNo (eps_ 1) x0 leaving 4 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying H2.
The subproof is completed by applying L5.
The subproof is completed by applying H4.