Let x0 of type ι → (ι → (ι → ο) → ο) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Apply H0 with
λ x3 . λ x4 : (ι → ο) → ο . ∃ x5 : ι → (ι → ο) → ο . and (∀ x6 . x6 ∈ x3 ⟶ 2b1c2.. x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ι → (ι → ο) → ο be given.
Assume H1:
∀ x5 . x5 ∈ x3 ⟶ ∃ x6 : ι → (ι → ο) → ο . and (∀ x7 . x7 ∈ x5 ⟶ 2b1c2.. x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2:
∀ x6 : ι → (ι → ο) → ο . and (∀ x7 . x7 ∈ x3 ⟶ 2b1c2.. x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6) ⟶ x5.
Apply H2 with
x4.
Apply andI with
∀ x6 . x6 ∈ x3 ⟶ 2b1c2.. x0 x6 (x4 x6),
x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Assume H3: x6 ∈ x3.
Apply H1 with
x6,
2b1c2.. x0 x6 (x4 x6) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x7 of type ι → (ι → ο) → ο be given.
Assume H4:
and (∀ x8 . x8 ∈ x6 ⟶ 2b1c2.. x0 x8 (x7 x8)) (x4 x6 = x0 x6 x7).
Apply H4 with
2b1c2.. x0 x6 (x4 x6).
Assume H5:
∀ x8 . x8 ∈ x6 ⟶ 2b1c2.. x0 x8 (x7 x8).
Assume H6: x4 x6 = x0 x6 x7.
Apply H6 with
λ x8 x9 : (ι → ο) → ο . 2b1c2.. x0 x6 x9.
Apply unknownprop_4827c395f66ed5e2b9711333b1b1681a1ac1c67519cd7de68052d7f43b6fa12f with
x0,
x6,
x7.
The subproof is completed by applying H5.
Let x6 of type ((ι → ο) → ο) → ((ι → ο) → ο) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.