Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: setsum 0 x2 ∈ setsum x0 x1.

Apply pairE with x0, x1, setsum 0 x2, x2 ∈ x0 leaving 3 subgoals.

The subproof is completed by applying H0.

Apply Inj0_pair_0_eq with λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x0) (x3 x2 = x3 x5)) ⟶ x2 ∈ x0.

Assume H1: ∃ x3 . and (x3 ∈ x0) (Inj0 x2 = Inj0 x3).

Apply exandE_i with λ x3 . x3 ∈ x0, λ x3 . Inj0 x2 = Inj0 x3, x2 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H1.

Let x3 of type ι be given.

Assume H2: x3 ∈ x0.

Assume H3: Inj0 x2 = Inj0 x3.

Apply Inj0_inj with x2, x3, λ x4 x5 . x5 ∈ x0 leaving 2 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H2.

Apply Inj0_pair_0_eq with λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x1) (x3 x2 = setsum 1 x5)) ⟶ x2 ∈ x0.

Apply Inj1_pair_1_eq with λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x1) (Inj0 x2 = x3 x5)) ⟶ x2 ∈ x0.

Assume H1: ∃ x3 . and (x3 ∈ x1) (Inj0 x2 = Inj1 x3).

Apply FalseE with x2 ∈ x0.

Apply exandE_i with λ x3 . x3 ∈ x1, λ x3 . Inj0 x2 = Inj1 x3, False leaving 2 subgoals.

The subproof is completed by applying H1.

Let x3 of type ι be given.

Assume H2: x3 ∈ x1.

Assume H3: Inj0 x2 = Inj1 x3.

Apply Inj0_Inj1_neq with x2, x3.

The subproof is completed by applying H3.

■

Proofgold Proof