Let x0 of type ι be given.
Let x1 of type ι be given.
Apply set_ext with
setsum x0 x1,
lam 2 (λ x2 . If_i (x2 = 0) x0 x1) leaving 2 subgoals.
Let x2 of type ι be given.
Apply pairE with
x0,
x1,
x2,
x2 ∈ lam 2 (λ x3 . If_i (x3 = 0) x0 x1) leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1:
∃ x3 . and (x3 ∈ x0) (x2 = setsum 0 x3).
Apply exandE_i with
λ x3 . x3 ∈ x0,
λ x3 . x2 = setsum 0 x3,
x2 ∈ lam 2 (λ x3 . If_i (x3 = 0) x0 x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Apply H3 with
λ x4 x5 . x5 ∈ lam 2 (λ x6 . If_i (x6 = 0) x0 x1).
Apply lamI with
2,
λ x4 . If_i (x4 = 0) x0 x1,
0,
x3 leaving 2 subgoals.
The subproof is completed by applying In_0_2.
Apply If_i_1 with
0 = 0,
x0,
x1,
λ x4 x5 . x3 ∈ x5 leaving 2 subgoals.
Let x4 of type ι → ι → ο be given.
Assume H4: x4 0 0.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
Assume H1:
∃ x3 . and (x3 ∈ x1) (x2 = setsum 1 x3).
Apply exandE_i with
λ x3 . x3 ∈ x1,
λ x3 . x2 = setsum 1 x3,
x2 ∈ lam 2 (λ x3 . If_i (x3 = 0) x0 x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x1.
Apply H3 with
λ x4 x5 . x5 ∈ lam 2 (λ x6 . If_i (x6 = 0) x0 x1).
Apply lamI with
2,
λ x4 . If_i (x4 = 0) x0 x1,
1,
x3 leaving 2 subgoals.
The subproof is completed by applying In_1_2.
Apply If_i_0 with
1 = 0,
x0,
x1,
λ x4 x5 . x3 ∈ x5 leaving 2 subgoals.
The subproof is completed by applying neq_1_0.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H0:
x2 ∈ lam 2 (λ x3 . If_i (x3 = 0) x0 x1).
Apply exandE_i with
λ x3 . x3 ∈ 2,
λ x3 . ∃ x4 . and (x4 ∈ If_i (x3 = 0) x0 x1) (x2 = setsum x3 x4),
x2 ∈ setsum x0 x1 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,
x2 ∈ setsum x0 x1 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.
Apply H5 with
λ x5 x6 . x6 ∈ ....