Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1.
Let x5 of type ι be given.
Assume H2: x5 ∈ x1.
Apply H0 with
λ x6 x7 . x2 x4 x5 = decode_r (ap x7 1) x4 x5.
Apply tuple_3_1_eq with
x1,
encode_r x1 x2,
x3,
λ x6 x7 . x2 x4 x5 = decode_r x7 x4 x5.
Let x6 of type ο → ο → ο be given.
Apply decode_encode_r with
x1,
x2,
x4,
x5,
λ x7 x8 : ο . x6 x8 x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.