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