Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
In x2 (Pi x0 (λ x3 . x1 x3)).
Let x3 of type ι be given.
Assume H1:
In x3 (Pi x0 (λ x4 . x1 x4)).
Assume H2:
∀ x4 . In x4 x0 ⟶ ap x2 x4 = ap x3 x4.
Apply unknownprop_a23ec6a55ac212526d74cbf0d04096929ad453b0eb0f8023e32b8a33930d39fb with
x2,
x3 leaving 2 subgoals.
Apply unknownprop_9684b75224577c1a9f3ef932f37b496157d36cc8bb2bcf04fcf5f33afc9a9e8c with
x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Apply H2 with
x4,
λ x5 x6 . Subq x6 (ap x3 x4) leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying unknownprop_d889823a5c975ad2d68f484964233a1e69e7d67f0888aa5b83d962eeca107acd with
ap x3 x4.
Apply unknownprop_9684b75224577c1a9f3ef932f37b496157d36cc8bb2bcf04fcf5f33afc9a9e8c with
x0,
x1,
x3,
x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Apply H2 with
x4,
λ x5 x6 . Subq (ap x3 x4) x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying unknownprop_d889823a5c975ad2d68f484964233a1e69e7d67f0888aa5b83d962eeca107acd with
ap x3 x4.