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.
Let x4 of type ι be given.
Assume H0:
x4 ∈ lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3))).
Claim L1:
∃ x5 . and (x5 ∈ 4) (∃ x6 . and (x6 ∈ If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3))) (x4 = setsum x5 x6))Apply lamE with
4,
λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)),
x4.
The subproof is completed by applying H0.
Apply exandE_i with
λ x5 . x5 ∈ 4,
λ x5 . ∃ x6 . and (x6 ∈ If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3))) (x4 = setsum x5 x6),
∃ x5 . and (x5 ∈ 4) (∃ x6 . x4 = setsum x5 x6) leaving 2 subgoals.
The subproof is completed by applying L1.
Let x5 of type ι be given.
Assume H2: x5 ∈ 4.
Assume H3:
∃ x6 . and (x6 ∈ If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3))) (x4 = setsum x5 x6).
Apply exandE_i with
λ x6 . x6 ∈ If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)),
λ x6 . x4 = setsum x5 x6,
∃ x6 . and (x6 ∈ 4) (∃ x7 . x4 = setsum x6 x7) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x6 of type ι be given.
Assume H4:
x6 ∈ If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)).
Let x7 of type ο be given.
Assume H6:
∀ x8 . and (x8 ∈ 4) (∃ x9 . x4 = setsum x8 x9) ⟶ x7.
Apply H6 with
x5.
Apply andI with
x5 ∈ 4,
∃ x8 . x4 = setsum x5 x8 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x8 of type ο be given.
Assume H7:
∀ x9 . x4 = setsum x5 x9 ⟶ x8.
Apply H7 with
x6.
The subproof is completed by applying H5.