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: x0 ⊆ x2.
Assume H1: x1 ⊆ x3.
Let x4 of type ι be given.
Apply setsum_Inj_inv with
x0,
x1,
x4,
x4 ∈ setsum x2 x3 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3:
∃ x5 . and (x5 ∈ x0) (x4 = Inj0 x5).
Apply exandE_i with
λ x5 . x5 ∈ x0,
λ x5 . x4 = Inj0 x5,
x4 ∈ setsum x2 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Apply H5 with
λ x6 x7 . x7 ∈ setsum x2 x3.
Apply Inj0_setsum with
x2,
x3,
x5.
Apply H0 with
x5.
The subproof is completed by applying H4.
Assume H3:
∃ x5 . and (x5 ∈ x1) (x4 = Inj1 x5).
Apply exandE_i with
λ x5 . x5 ∈ x1,
λ x5 . x4 = Inj1 x5,
x4 ∈ setsum x2 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x5 of type ι be given.
Assume H4: x5 ∈ x1.
Apply H5 with
λ x6 x7 . x7 ∈ setsum x2 x3.
Apply Inj1_setsum with
x2,
x3,
x5.
Apply H1 with
x5.
The subproof is completed by applying H4.