Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply mul_SNo_oneR with
x2,
λ x3 x4 . SNoLt (div_SNo x0 x1) x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply recip_SNo_invR with
x1,
λ x3 x4 . SNoLt (div_SNo x0 x1) (mul_SNo x2 x3) leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H5: x1 = 0.
Apply SNoLt_irref with
x1.
Apply H5 with
λ x3 x4 . SNoLt x4 x1.
The subproof is completed by applying H3.
Apply mul_SNo_assoc with
x2,
x1,
recip_SNo x1,
λ x3 x4 . SNoLt (mul_SNo x0 (recip_SNo x1)) x4 leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_recip_SNo with
x1.
The subproof is completed by applying H1.
Apply pos_mul_SNo_Lt' with
x0,
mul_SNo x2 x1,
recip_SNo x1 leaving 5 subgoals.
The subproof is completed by applying H0.
Apply SNo_mul_SNo with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Apply SNo_recip_SNo with
x1.
The subproof is completed by applying H1.
Apply recip_SNo_of_pos_is_pos with
x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.