Let x0 of type ι → (ι → (((ι → ο) → ο) → ο) → ο) → (((ι → ο) → ο) → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type (((ι → ο) → ο) → ο) → ο be given.
Apply H0 with
λ x3 . λ x4 : (((ι → ο) → ο) → ο) → ο . ∃ x5 : ι → (((ι → ο) → ο) → ο) → ο . and (∀ x6 . prim1 x6 x3 ⟶ 9eb9b.. x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ι → (((ι → ο) → ο) → ο) → ο be given.
Assume H1:
∀ x5 . prim1 x5 x3 ⟶ ∃ x6 : ι → (((ι → ο) → ο) → ο) → ο . and (∀ x7 . prim1 x7 x5 ⟶ 9eb9b.. x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2:
∀ x6 : ι → (((ι → ο) → ο) → ο) → ο . and (∀ x7 . prim1 x7 x3 ⟶ 9eb9b.. x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6) ⟶ x5.
Apply H2 with
x4.
Apply andI with
∀ x6 . prim1 x6 x3 ⟶ 9eb9b.. x0 x6 (x4 x6),
x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Apply H1 with
x6,
9eb9b.. x0 x6 (x4 x6) leaving 2 subgoals.
The subproof is completed by applying H3.
Let x7 of type ι → (((ι → ο) → ο) → ο) → ο be given.
Assume H4:
and (∀ x8 . prim1 x8 x6 ⟶ 9eb9b.. x0 x8 (x7 x8)) (x4 x6 = x0 x6 x7).
Apply H4 with
9eb9b.. x0 x6 (x4 x6).
Assume H6: x4 x6 = x0 x6 x7.
Apply H6 with
λ x8 x9 : (((ι → ο) → ο) → ο) → ο . 9eb9b.. x0 x6 x9.
Apply unknownprop_baa64b0dc3150faa2aa910b1d0b43f6e89eea54fa7fdbb099eecf828a22d69a1 with
x0,
x6,
x7.
The subproof is completed by applying H5.
Let x6 of type ((((ι → ο) → ο) → ο) → ο) → ((((ι → ο) → ο) → ο) → ο) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.