Let x0 of type ι be given.
Let x1 of type ι be given.
Apply unknownprop_c86200a2eefb0ff844f50b29d5cbeaa2ee14856a2db63542bcbf63218f0d0f1e with
x0,
λ x2 . PtdPred (f5637.. x2 x1) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Assume H2: x4 ∈ x2.
Assume H3: x3 x4.
Apply unknownprop_c86200a2eefb0ff844f50b29d5cbeaa2ee14856a2db63542bcbf63218f0d0f1e with
x1,
λ x5 . PtdPred (f5637.. (pack_p x2 x3) x5) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x5 of type ι be given.
Let x6 of type ι → ο be given.
Let x7 of type ι be given.
Assume H4: x7 ∈ x5.
Assume H5: x6 x7.
Apply unknownprop_8a66e4bdb214ee413447ac05cf120a4276b1afb59a2fc210ec5bfee37471a43f with
x2,
x3,
x5,
x6,
λ x8 x9 . PtdPred x9.
Apply unknownprop_2576d2815b46016e5e13a9989b4e9789629d83c56ed1c92a4cda0de077a20752 with
setexp x5 x2,
λ x8 . ∀ x9 . x9 ∈ x2 ⟶ x3 x9 ⟶ x6 (ap x8 x9).
Let x8 of type ο be given.
Assume H6:
∀ x9 . and (x9 ∈ setexp x5 x2) ((λ x10 . ∀ x11 . x11 ∈ x2 ⟶ x3 x11 ⟶ x6 (ap x10 x11)) x9) ⟶ x8.
Apply H6 with
lam x2 (λ x9 . x7).
Apply andI with
lam x2 (λ x9 . x7) ∈ setexp x5 x2,
(λ x9 . ∀ x10 . x10 ∈ x2 ⟶ x3 x10 ⟶ x6 (ap x9 x10)) (lam x2 (λ x9 . x7)) leaving 2 subgoals.
Apply lam_Pi with
x2,
λ x9 . x5,
λ x9 . x7.
Let x9 of type ι be given.
Assume H7: x9 ∈ x2.
The subproof is completed by applying H4.
Let x9 of type ι be given.
Assume H7: x9 ∈ x2.
Assume H8: x3 x9.
Claim L9:
ap (lam x2 (λ x10 . x7)) x9 = x7Apply beta with
x2,
λ x10 . x7,
x9.
The subproof is completed by applying H7.
Apply L9 with
λ x10 x11 . x6 x11.
The subproof is completed by applying H5.