Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Apply iffI with
x2 ∈ lam x0 (λ x3 . x1 x3),
∃ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x1 x3) (x2 = setsum x3 x4)) leaving 2 subgoals.
The subproof is completed by applying lamE with x0, λ x3 . x1 x3, x2.
Assume H0:
∃ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x1 x3) (x2 = setsum x3 x4)).
Apply exandE_i with
λ x3 . x3 ∈ x0,
λ x3 . ∃ x4 . and (x4 ∈ x1 x3) (x2 = setsum x3 x4),
x2 ∈ lam x0 (λ x3 . x1 x3) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Assume H2:
∃ x4 . and (x4 ∈ x1 x3) (x2 = setsum x3 x4).
Apply exandE_i with
λ x4 . x4 ∈ x1 x3,
λ x4 . x2 = setsum x3 x4,
x2 ∈ lam x0 (λ x4 . x1 x4) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x4 of type ι be given.
Assume H3: x4 ∈ x1 x3.
Apply H4 with
λ x5 x6 . x6 ∈ lam x0 (λ x7 . x1 x7).
Apply lamI with
x0,
λ x5 . x1 x5,
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.