Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ lam x0 (λ x3 . x1 x3).
Apply and3E with
setsum (proj0 x2) (proj1 x2) = x2,
proj0 x2 ∈ x0,
proj1 x2 ∈ x1 (proj0 x2),
∃ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x1 x3) (x2 = setsum x3 x4)) leaving 2 subgoals.
Apply Sigma_eta_proj0_proj1 with
x0,
x1,
x2.
The subproof is completed by applying H0.
Assume H2:
proj0 x2 ∈ x0.
Let x3 of type ο be given.
Assume H4:
∀ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x1 x4) (x2 = setsum x4 x5)) ⟶ x3.
Apply H4 with
proj0 x2.
Apply andI with
proj0 x2 ∈ x0,
∃ x4 . and (x4 ∈ x1 (proj0 x2)) (x2 = setsum (proj0 x2) x4) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x4 of type ο be given.
Apply H5 with
proj1 x2.
Apply andI with
proj1 x2 ∈ x1 (proj0 x2),
x2 = setsum (proj0 x2) (proj1 x2) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι → ι → ο be given.
The subproof is completed by applying H1 with λ x6 x7 . x5 x7 x6.