Let x0 of type ι → ο be given.
Assume H1:
x0 (pack_u 1 (λ x1 . x1)).
Apply andI with
x0 (pack_u 1 (λ x1 . x1)),
∀ x1 . x0 x1 ⟶ and (UnaryFuncHom x1 (pack_u 1 (λ x2 . x2)) ((λ x2 . lam (ap x2 0) (λ x3 . 0)) x1)) (∀ x2 . UnaryFuncHom x1 (pack_u 1 (λ x3 . x3)) x2 ⟶ x2 = (λ x3 . lam (ap x3 0) (λ x4 . 0)) x1) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H2: x0 x1.
Apply H0 with
x1,
λ x2 . and (UnaryFuncHom x2 (pack_u 1 (λ x3 . x3)) ((λ x3 . lam (ap x3 0) (λ x4 . 0)) x2)) (∀ x3 . UnaryFuncHom x2 (pack_u 1 (λ x4 . x4)) x3 ⟶ x3 = (λ x4 . lam (ap x4 0) (λ x5 . 0)) x2) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Let x3 of type ι → ι be given.
Assume H3: ∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2.
Apply andI with
UnaryFuncHom (pack_u x2 x3) (pack_u 1 (λ x4 . x4)) ((λ x4 . lam (ap x4 0) (λ x5 . 0)) (pack_u x2 x3)),
∀ x4 . UnaryFuncHom (pack_u x2 x3) (pack_u 1 (λ x5 . x5)) x4 ⟶ x4 = (λ x5 . lam (ap x5 0) (λ x6 . 0)) (pack_u x2 x3) leaving 2 subgoals.
Apply pack_u_0_eq2 with
x2,
x3,
λ x4 x5 . UnaryFuncHom (pack_u x2 x3) (pack_u 1 (λ x6 . x6)) (lam x4 (λ x6 . 0)).
Apply unknownprop_c0506b7ce99ca057359584255bdeac2239c78bf84c4390e2fc4c72ca99c22cfa with
x2,
1,
x3,
λ x4 . x4,
lam x2 (λ x4 . 0),
λ x4 x5 : ο . x5.
Apply andI with
lam x2 (λ x4 . 0) ∈ setexp 1 x2,
∀ x4 . x4 ∈ x2 ⟶ ap (lam x2 (λ x5 . 0)) (x3 x4) = (λ x5 . x5) (ap (lam x2 (λ x5 . 0)) x4) leaving 2 subgoals.
Apply lam_Pi with
x2,
λ x4 . 1,
λ x4 . 0.
Let x4 of type ι be given.
Assume H4: x4 ∈ x2.
The subproof is completed by applying In_0_1.
Let x4 of type ι be given.
Assume H4: x4 ∈ x2.
Apply beta with
x2,
λ x5 . 0,
x4,
λ x5 x6 . ap (lam x2 (λ x7 . 0)) (x3 x4) = x6 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply beta with
x2,
λ x5 . 0,
x3 x4.
Apply H3 with
x4.
The subproof is completed by applying H4.
Let x4 of type ι be given.
Apply unknownprop_c0506b7ce99ca057359584255bdeac2239c78bf84c4390e2fc4c72ca99c22cfa with
x2,
1,
x3,
λ x5 . x5,
x4,
λ x5 x6 : ο . x6 ⟶ x4 = (λ x7 . lam (ap x7 0) (λ x8 . 0)) (pack_u x2 x3).
Assume H4:
and (x4 ∈ setexp 1 x2) (∀ x5 . ... ⟶ ap x4 (x3 x5) = (λ x6 . x6) (ap x4 x5)).