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 H0: x0 x2 x2.
Apply If_i_correct with
x0 x2 (x1 x2),
x1 x2,
canonical_elt x0 x2,
x0 x2 (If_i (x0 x2 (x1 x2)) (x1 x2) (canonical_elt x0 x2)) leaving 2 subgoals.
Apply H1 with
x0 x2 (If_i (x0 x2 (x1 x2)) (x1 x2) (canonical_elt x0 x2)).
Assume H2: x0 x2 (x1 x2).
Apply H3 with
λ x3 x4 . x0 x2 x4.
The subproof is completed by applying H2.
Apply andER with
not (x0 x2 (x1 x2)),
If_i (x0 x2 (x1 x2)) (x1 x2) (canonical_elt x0 x2) = canonical_elt x0 x2,
λ x3 x4 . x0 x2 x4 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply canonical_elt_rel with
x0,
x2.
The subproof is completed by applying H0.