Let x0 of type ι → (ι → ι → ο) → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Apply H0 with
λ x3 . λ x4 : ι → ο . ∃ x5 : ι → ι → ο . and (∀ x6 . In x6 x3 ⟶ 6869c.. x0 x6 (x5 x6)) (x4 = x0 x3 x5).
Let x3 of type ι be given.
Let x4 of type ι → ι → ο be given.
Assume H1:
∀ x5 . In x5 x3 ⟶ ∃ x6 : ι → ι → ο . and (∀ x7 . In x7 x5 ⟶ 6869c.. x0 x7 (x6 x7)) (x4 x5 = x0 x5 x6).
Let x5 of type ο be given.
Assume H2:
∀ x6 : ι → ι → ο . and (∀ x7 . In x7 x3 ⟶ 6869c.. x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6) ⟶ x5.
Apply H2 with
x4.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
∀ x6 . In x6 x3 ⟶ 6869c.. x0 x6 (x4 x6),
x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Apply unknownprop_98b9aeda084485fc944a7c4f01aecf8f7bcf5e349b4c35ef0e56e9fe7caff0eb with
λ x7 : ι → ι → ο . ∀ x8 . In x8 x6 ⟶ 6869c.. x0 x8 (x7 x8),
λ x7 : ι → ι → ο . x4 x6 = x0 x6 x7,
6869c.. x0 x6 (x4 x6) leaving 2 subgoals.
Apply H1 with
x6.
The subproof is completed by applying H3.
Let x7 of type ι → ι → ο be given.
Assume H4:
∀ x8 . In x8 x6 ⟶ 6869c.. x0 x8 (x7 x8).
Assume H5: x4 x6 = x0 x6 x7.
Apply H5 with
λ x8 x9 : ι → ο . 6869c.. x0 x6 x9.
Apply unknownprop_7e1433f0d7534eb35553ba0ab0a1f0e9ad42c403c2045d379ba59305b2ddd888 with
x0,
x6,
x7.
The subproof is completed by applying H4.
Let x6 of type (ι → ο) → (ι → ο) → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.