Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H0:
x3 ∈ lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)).
Claim L1:
∃ x4 . and (x4 ∈ 3) (∃ x5 . and (x5 ∈ If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)) (x3 = setsum x4 x5))Apply lamE with
3,
λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2),
x3.
The subproof is completed by applying H0.
Apply exandE_i with
λ x4 . x4 ∈ 3,
λ x4 . ∃ x5 . and (x5 ∈ If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)) (x3 = setsum x4 x5),
∃ x4 . and (x4 ∈ 3) (∃ x5 . x3 = setsum x4 x5) leaving 2 subgoals.
The subproof is completed by applying L1.
Let x4 of type ι be given.
Assume H2: x4 ∈ 3.
Assume H3:
∃ x5 . and (x5 ∈ If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)) (x3 = setsum x4 x5).
Apply exandE_i with
λ x5 . x5 ∈ If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2),
λ x5 . x3 = setsum x4 x5,
∃ x5 . and (x5 ∈ 3) (∃ x6 . x3 = setsum x5 x6) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4:
x5 ∈ If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2).
Let x6 of type ο be given.
Assume H6:
∀ x7 . and (x7 ∈ 3) (∃ x8 . x3 = setsum x7 x8) ⟶ x6.
Apply H6 with
x4.
Apply andI with
x4 ∈ 3,
∃ x7 . x3 = setsum x4 x7 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x7 of type ο be given.
Assume H7:
∀ x8 . x3 = setsum x4 x8 ⟶ x7.
Apply H7 with
x5.
The subproof is completed by applying H5.