Let x0 of type ι → ι → ο be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H1: x0 x2 (x1 x2).
Assume H2: x2 = x1 x2.
Apply andI with
x0 x2 x2,
x2 = canonical_elt_def x0 x1 x2 leaving 2 subgoals.
Apply per_ref1 with
x0,
x2,
x1 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply If_i_1 with
x0 x2 (x1 x2),
x1 x2,
canonical_elt x0 x2,
λ x3 x4 . x2 = x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.