Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoLt_trichotomy_or_impred with
0,
x1,
SNoLt 0 x1 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Assume H4: 0 = x1.
Apply FalseE with
SNoLt 0 x1.
Apply neq_0_1.
Apply H3 with
λ x2 x3 . 0 = x2.
Apply H4 with
λ x2 x3 . 0 = mul_SNo x0 x2.
Let x2 of type ι → ι → ο be given.
Apply mul_SNo_zeroR with
x0,
λ x3 x4 . x2 x4 x3.
The subproof is completed by applying H0.
Apply FalseE with
SNoLt 0 x1.
Apply SNoLt_irref with
0.
Apply SNoLt_tra with
0,
1,
0 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying SNo_1.
The subproof is completed by applying SNo_0.
The subproof is completed by applying SNoLt_0_1.
Apply H3 with
λ x2 x3 . SNoLt x2 0.
Apply mul_SNo_pos_neg with
x0,
x1 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.