Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Assume H0:
∀ x3 . x3 ∈ x0 ⟶ x2 (Inj0 x3).
Assume H1:
∀ x3 . x3 ∈ x1 ⟶ x2 (Inj1 x3).
Let x3 of type ι be given.
Apply setsum_Inj_inv with
x0,
x1,
x3,
x2 x3 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3:
∃ x4 . and (x4 ∈ x0) (x3 = Inj0 x4).
Apply H3 with
x2 x3.
Let x4 of type ι be given.
Assume H4:
(λ x5 . and (x5 ∈ x0) (x3 = Inj0 x5)) x4.
Apply H4 with
x2 x3.
Assume H5: x4 ∈ x0.
Apply H6 with
λ x5 x6 . x2 x6.
Apply H0 with
x4.
The subproof is completed by applying H5.
Assume H3:
∃ x4 . and (x4 ∈ x1) (x3 = Inj1 x4).
Apply H3 with
x2 x3.
Let x4 of type ι be given.
Assume H4:
(λ x5 . and (x5 ∈ x1) (x3 = Inj1 x5)) x4.
Apply H4 with
x2 x3.
Assume H5: x4 ∈ x1.
Apply H6 with
λ x5 x6 . x2 x6.
Apply H1 with
x4.
The subproof is completed by applying H5.