Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι be given.
Assume H0: x3 ∈ x0.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1 x3.
Apply beta with
x0,
λ x5 . lam (x1 x5) (λ x6 . x2 x5 x6),
x3,
λ x5 x6 . ap x6 x4 = x2 x3 x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply beta with
x1 x3,
λ x5 . x2 x3 x5,
x4.
The subproof is completed by applying H1.