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: equip x0 x2.

Assume H1: equip x1 x3.

Assume H2: ∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1.

Apply H0 with equip (binunion x0 x1) (setsum x2 x3).

Let x4 of type ι → ι be given.

Assume H3: bij x0 x2 x4.

Apply bijE with x0, x2, x4, equip (binunion x0 x1) (setsum x2 x3) leaving 2 subgoals.

The subproof is completed by applying H3.

Assume H4: ∀ x5 . x5 ∈ x0 ⟶ x4 x5 ∈ x2.

Assume H5: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 = x4 x6 ⟶ x5 = x6.

Assume H6: ∀ x5 . x5 ∈ x2 ⟶ ∃ x6 . and (x6 ∈ x0) (x4 x6 = x5).

Apply H1 with equip (binunion x0 x1) (setsum x2 x3).

Let x5 of type ι → ι be given.

Assume H7: bij x1 x3 x5.

Apply bijE with x1, x3, x5, equip (binunion x0 x1) (setsum x2 x3) leaving 2 subgoals.

The subproof is completed by applying H7.

Assume H8: ∀ x6 . x6 ∈ x1 ⟶ x5 x6 ∈ x3.

Assume H9: ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7.

Assume H10: ∀ x6 . x6 ∈ x3 ⟶ ∃ x7 . and (x7 ∈ x1) (x5 x7 = x6).

Let x6 of type ο be given.

Assume H11: ∀ x7 : ι → ι . bij (binunion x0 x1) (setsum x2 x3) x7 ⟶ x6.

Apply H11 with λ x7 . If_i (x7 ∈ x0) (Inj0 (x4 x7)) (Inj1 (x5 x7)).

Apply bijI with binunion x0 x1, setsum x2 x3, λ x7 . If_i (x7 ∈ x0) (Inj0 (x4 x7)) (Inj1 (x5 x7)) leaving 3 subgoals.

Let x7 of type ι be given.

Assume H12: x7 ∈ binunion x0 x1.

Apply binunionE with x0, x1, x7, (λ x8 . If_i (x8 ∈ x0) (Inj0 (x4 x8)) (Inj1 (x5 x8))) x7 ∈ setsum x2 x3 leaving 3 subgoals.

The subproof is completed by applying H12.

Assume H13: x7 ∈ x0.

Apply If_i_1 with x7 ∈ x0, Inj0 (x4 x7), Inj1 (x5 x7), λ x8 x9 . x9 ∈ setsum x2 x3 leaving 2 subgoals.

The subproof is completed by applying H13.

Apply Inj0_setsum with x2, x3, x4 x7.

Apply H4 with x7.

The subproof is completed by applying H13.

Assume H13: x7 ∈ x1.

Claim L14: nIn x7 x0

Assume H14: x7 ∈ x0.

Apply H2 with x7 leaving 2 subgoals.

The subproof is completed by applying H14.

The subproof is completed by applying H13.

Apply If_i_0 with x7 ∈ x0, Inj0 (x4 x7), Inj1 (x5 x7), λ x8 x9 . x9 ∈ setsum x2 x3 leaving 2 subgoals.

The subproof is completed by applying L14.

Apply Inj1_setsum with x2, x3, x5 x7.

Apply H8 with x7.

The subproof is completed by applying H13.

Let x7 of type ι be given.

Assume H12: x7 ∈ binunion x0 x1.

Let x8 of type ι be given.

Assume H13: x8 ∈ binunion x0 x1.

Apply binunionE with x0, ..., ..., ... leaving 3 subgoals.

...

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Proofgold Proof