Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoLt_trichotomy_or with
x0,
x1,
or (SNoLt x0 x1) (SNoLe x1 x0) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H2:
or (SNoLt x0 x1) (x0 = x1).
Apply H2 with
or (SNoLt x0 x1) (SNoLe x1 x0) leaving 2 subgoals.
Apply orIL with
SNoLt x0 x1,
SNoLe x1 x0.
The subproof is completed by applying H3.
Assume H3: x0 = x1.
Apply orIR with
SNoLt x0 x1,
SNoLe x1 x0.
Apply H3 with
λ x2 x3 . SNoLe x1 x3.
The subproof is completed by applying SNoLe_ref with x1.
Apply orIR with
SNoLt x0 x1,
SNoLe x1 x0.
Apply SNoLtLe with
x1,
x0.
The subproof is completed by applying H2.