Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ο be given.
Assume H2:
prim1 x0 x1 ⟶ x2.
Assume H3: x0 = x1 ⟶ x2.
Assume H4:
prim1 x1 x0 ⟶ x2.
Apply ordinal_trichotomy_or with
x0,
x1,
x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H5:
or (prim1 x0 x1) (x0 = x1).
Apply H5 with
x2 leaving 2 subgoals.
Apply H2.
The subproof is completed by applying H6.
Assume H6: x0 = x1.
Apply H3.
The subproof is completed by applying H6.
Apply H4.
The subproof is completed by applying H5.