Let x0 of type ι → ο be given.
Apply unknownprop_1db1571afe8c01990252b7801041a0001ba1fedff9d78947d027d61a0ff0ae7f with
x0,
λ x1 . ap x1 0,
UnaryFuncHom leaving 3 subgoals.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x0 x1.
Assume H2: x0 x2.
Apply H0 with
x1,
λ x4 . UnaryFuncHom x4 x2 x3 ⟶ x3 ∈ setexp (ap x2 0) (ap x4 0) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Let x5 of type ι → ι be given.
Assume H3: ∀ x6 . x6 ∈ x4 ⟶ x5 x6 ∈ x4.
Apply H0 with
x2,
λ x6 . UnaryFuncHom (pack_u x4 x5) x6 x3 ⟶ x3 ∈ setexp (ap x6 0) (ap (pack_u x4 x5) 0) leaving 2 subgoals.
The subproof is completed by applying H2.
Let x6 of type ι be given.
Let x7 of type ι → ι be given.
Assume H4: ∀ x8 . x8 ∈ x6 ⟶ x7 x8 ∈ x6.
Apply unknownprop_c0506b7ce99ca057359584255bdeac2239c78bf84c4390e2fc4c72ca99c22cfa with
x4,
x6,
x5,
x7,
x3,
λ x8 x9 : ο . x9 ⟶ x3 ∈ setexp (ap (pack_u x6 x7) 0) (ap (pack_u x4 x5) 0).
Assume H5:
and (x3 ∈ setexp x6 x4) (∀ x8 . x8 ∈ x4 ⟶ ap x3 (x5 x8) = x7 (ap x3 x8)).
Apply H5 with
x3 ∈ setexp (ap (pack_u x6 x7) 0) (ap (pack_u x4 x5) 0).
Assume H7:
∀ x8 . x8 ∈ x4 ⟶ ap x3 (x5 x8) = x7 (ap x3 x8).
Apply pack_u_0_eq2 with
x6,
x7,
λ x8 x9 . x3 ∈ setexp x8 (ap (pack_u x4 x5) 0).
Apply pack_u_0_eq2 with
x4,
x5,
λ x8 x9 . x3 ∈ setexp x6 x8.
The subproof is completed by applying H6.
Let x1 of type ι be given.
Assume H1: x0 x1.
Apply H0 with
x1,
λ x2 . UnaryFuncHom x2 x2 (lam_id (ap x2 0)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Let x3 of type ι → ι be given.
Assume H2: ∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2.
Apply pack_u_0_eq2 with
x2,
x3,
λ x4 x5 . UnaryFuncHom (pack_u x2 x3) (pack_u x2 x3) (lam_id x4).
Apply unknownprop_c0506b7ce99ca057359584255bdeac2239c78bf84c4390e2fc4c72ca99c22cfa with
x2,
x2,
x3,
x3,
lam_id x2,
λ x4 x5 : ο . x5.
Apply andI with
lam_id x2 ∈ setexp x2 x2,
∀ x4 . x4 ∈ x2 ⟶ ap (lam_id x2) (x3 x4) = x3 (ap (lam_id x2) x4) leaving 2 subgoals.
The subproof is completed by applying lam_id_exp_In with x2.
Let x4 of type ι be given.
Assume H3: x4 ∈ x2.
Apply beta with
x2,
λ x5 . x5,
x4,
λ x5 x6 . ap (lam_id x2) (x3 x4) = x3 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply beta with
x2,
λ x5 . x5,
x3 x4.
Apply H2 with
x4.
The subproof is completed by applying H3.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H1: x0 x1.
Assume H2: x0 x2.
Assume H3: x0 x3.
Apply H0 with
x1,
λ x6 . ... ⟶ ... ⟶ UnaryFuncHom ... ... ... leaving 2 subgoals.