Let x0 of type ((ι → ο) → ο) → ο be given.
Let x1 of type ((ι → ο) → ο) → ο be given.
Apply functional extensionality with
x0,
x1.
Let x2 of type (ι → ο) → ο be given.
Apply prop_ext_2 with
x0 x2,
x1 x2 leaving 2 subgoals.
Assume H1: x0 x2.
Apply H0 with
λ x3 x4 : (((ι → ο) → ο) → ο) → ο . x3 (a4b00.. x2).
Let x3 of type ο be given.
Apply H2 with
x2.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
a4b00.. x2 = a4b00.. x2,
x0 x2 leaving 2 subgoals.
Let x4 of type (((ι → ο) → ο) → ο) → (((ι → ο) → ο) → ο) → ο be given.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
Apply L2 with
x1 x2.
Let x3 of type (ι → ο) → ο be given.
Apply andE with
a4b00.. x2 = a4b00.. x3,
x1 x3,
x1 x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H5: x1 x3.
Apply unknownprop_0baf4c86bc102cc7ef5b5a6fba0fc5f1dad77a50acb8dbafd746ed7c6f61a4d8 with
x2,
x3,
λ x4 x5 : (ι → ο) → ο . x1 x5 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H1: x1 x2.
Apply H0 with
λ x3 x4 : (((ι → ο) → ο) → ο) → ο . x4 (a4b00.. x2).
Let x3 of type ο be given.
Apply H2 with
x2.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
a4b00.. x2 = a4b00.. x2,
x1 x2 leaving 2 subgoals.
Let x4 of type (((ι → ο) → ο) → ο) → (((ι → ο) → ο) → ο) → ο be given.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
Apply L2 with
x0 x2.
Let x3 of type (ι → ο) → ο be given.
Apply andE with
a4b00.. x2 = a4b00.. x3,
x0 x3,
x0 x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H5: x0 x3.
Apply unknownprop_0baf4c86bc102cc7ef5b5a6fba0fc5f1dad77a50acb8dbafd746ed7c6f61a4d8 with
x2,
x3,
λ x4 x5 : (ι → ο) → ο . x0 x5 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.