Let x0 of type ι be given.
Apply H0 with
∃ x1 . and (x1 ∈ x0) (∃ x2 . and (x2 ∈ x0) (∃ x3 . and (x3 ∈ x0) (and (and (and (x1 = x2 ⟶ ∀ x4 : ο . x4) (x1 = x3 ⟶ ∀ x4 : ο . x4)) (x2 = x3 ⟶ ∀ x4 : ο . x4)) (∀ x4 . x4 ∈ x0 ⟶ or (or (x4 = x1) (x4 = x2)) (x4 = x3))))).
Let x1 of type ι → ι be given.
Apply H1 with
∃ x2 . and (x2 ∈ x0) (∃ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x0) (and (and (and (x2 = x3 ⟶ ∀ x5 : ο . x5) (x2 = x4 ⟶ ∀ x5 : ο . x5)) (x3 = x4 ⟶ ∀ x5 : ο . x5)) (∀ x5 . x5 ∈ x0 ⟶ or (or (x5 = x2) (x5 = x3)) (x5 = x4))))).
Assume H2:
and (∀ x2 . x2 ∈ 3 ⟶ x1 x2 ∈ x0) (∀ x2 . x2 ∈ 3 ⟶ ∀ x3 . x3 ∈ 3 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3).
Apply H2 with
(∀ x2 . x2 ∈ x0 ⟶ ∃ x3 . and (x3 ∈ 3) (x1 x3 = x2)) ⟶ ∃ x2 . and (x2 ∈ x0) (∃ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x0) (and (and (and (x2 = x3 ⟶ ∀ x5 : ο . x5) (x2 = x4 ⟶ ∀ x5 : ο . x5)) (x3 = x4 ⟶ ∀ x5 : ο . x5)) (∀ x5 . x5 ∈ x0 ⟶ or (or (x5 = x2) (x5 = x3)) (x5 = x4))))).
Assume H3: ∀ x2 . x2 ∈ 3 ⟶ x1 x2 ∈ x0.
Assume H4: ∀ x2 . x2 ∈ 3 ⟶ ∀ x3 . x3 ∈ 3 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Assume H5:
∀ x2 . x2 ∈ x0 ⟶ ∃ x3 . and (x3 ∈ 3) (x1 x3 = x2).
Let x2 of type ο be given.
Assume H6:
∀ x3 . and (x3 ∈ x0) (∃ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x0) (and (and (and (x3 = x4 ⟶ ∀ x6 : ο . x6) (x3 = x5 ⟶ ∀ x6 : ο . x6)) (x4 = x5 ⟶ ∀ x6 : ο . x6)) (∀ x6 . x6 ∈ x0 ⟶ or (or (x6 = x3) (x6 = x4)) (x6 = x5))))) ⟶ x2.