Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply setsum_Inj_inv with
x0,
x1,
x2,
Unj x2 ∈ x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H2:
∃ x3 . and (x3 ∈ x0) (x2 = Inj0 x3).
Apply H2 with
Unj x2 ∈ x1.
Let x3 of type ι be given.
Assume H3:
(λ x4 . and (x4 ∈ x0) (x2 = Inj0 x4)) x3.
Apply H3 with
Unj x2 ∈ x1.
Assume H4: x3 ∈ x0.
Apply FalseE with
Unj x2 ∈ x1.
Apply Inj0_Inj1_neq with
x3,
Unj x2.
Claim L6:
∀ x5 : ι → ο . x5 y4 ⟶ x5 (Inj0 x3)Let x5 of type ι → ο be given.
Apply H5 with
λ x6 x7 . (λ x8 . x5) x7 x6.
Apply H1 with
λ x6 . x5.
The subproof is completed by applying H6.
Let x5 of type ι → ι → ο be given.
Apply L6 with
λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H7: x5 y4 y4.
The subproof is completed by applying H7.
Assume H2:
∃ x3 . and (x3 ∈ x1) (x2 = Inj1 x3).
Apply H2 with
Unj x2 ∈ x1.
Let x3 of type ι be given.
Assume H3:
(λ x4 . and (x4 ∈ x1) (x2 = Inj1 x4)) x3.
Apply H3 with
Unj x2 ∈ x1.
Assume H4: x3 ∈ x1.
Apply H5 with
λ x4 x5 . Unj x5 ∈ x1.
Apply Unj_Inj1_eq with
x3,
λ x4 x5 . x5 ∈ x1.
The subproof is completed by applying H4.