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 H2:
∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1.
Apply H0 with
atleastp (binunion x0 x1) (setsum x2 x3).
Let x4 of type ι → ι be given.
Apply H3 with
atleastp (binunion x0 x1) (setsum x2 x3).
Assume H4: ∀ x5 . x5 ∈ x0 ⟶ x4 x5 ∈ x2.
Assume H5: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 = x4 x6 ⟶ x5 = x6.
Apply H1 with
atleastp (binunion x0 x1) (setsum x2 x3).
Let x5 of type ι → ι be given.
Apply H6 with
atleastp (binunion x0 x1) (setsum x2 x3).
Assume H7: ∀ x6 . x6 ∈ x1 ⟶ x5 x6 ∈ x3.
Assume H8: ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7.
Let x6 of type ο be given.
Apply H9 with
λ x7 . If_i (x7 ∈ x0) (Inj0 (x4 x7)) (Inj1 (x5 x7)).
Apply andI with
∀ x7 . x7 ∈ binunion x0 x1 ⟶ (λ x8 . If_i (x8 ∈ x0) (Inj0 (x4 x8)) (Inj1 (x5 x8))) x7 ∈ setsum x2 x3,
∀ x7 . x7 ∈ binunion x0 x1 ⟶ ∀ x8 . x8 ∈ binunion x0 x1 ⟶ (λ x9 . If_i (x9 ∈ x0) (Inj0 (x4 x9)) (Inj1 (x5 x9))) x7 = (λ x9 . If_i (x9 ∈ x0) (Inj0 (x4 x9)) (Inj1 (x5 x9))) x8 ⟶ x7 = x8 leaving 2 subgoals.
Let x7 of type ι be given.
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 H10.
Assume H11: 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 H11.
Apply Inj0_setsum with
x2,
x3,
x4 x7.
Apply H4 with
x7.
The subproof is completed by applying H11.
Assume H11: x7 ∈ x1.
Assume H12: x7 ∈ x0.
Apply H2 with
x7 leaving 2 subgoals.
The subproof is completed by applying H12.
The subproof is completed by applying H11.
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 L12.
Apply Inj1_setsum with
x2,
x3,
x5 x7.
Apply H7 with
x7.
The subproof is completed by applying H11.