Let x0 of type ι → ο be given.
Apply unknownprop_1db1571afe8c01990252b7801041a0001ba1fedff9d78947d027d61a0ff0ae7f with
x0,
λ x1 . ap x1 0,
PreContinuousHom 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 . PreContinuousHom 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 → ο) → ο be given.
Apply H0 with
x2,
λ x6 . PreContinuousHom (pack_c x4 x5) x6 x3 ⟶ x3 ∈ setexp (ap x6 0) (ap (pack_c 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 → ο) → ο be given.
Apply unknownprop_147946d52b6747e7a3735111f3622ca84b157f241b7b107aab3bab9bb651af48 with
x4,
x6,
x5,
x7,
x3,
λ x8 x9 : ο . x9 ⟶ x3 ∈ setexp (ap (pack_c x6 x7) 0) (ap (pack_c x4 x5) 0).
Assume H3:
and (x3 ∈ setexp x6 x4) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ x9 ∈ x6) ⟶ x7 x8 ⟶ x5 (λ x9 . and (x9 ∈ x4) (x8 (ap x3 x9)))).
Apply H3 with
x3 ∈ setexp (ap (pack_c x6 x7) 0) (ap (pack_c x4 x5) 0).
Assume H5:
∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ x9 ∈ x6) ⟶ x7 x8 ⟶ x5 (λ x9 . and (x9 ∈ x4) (x8 (ap x3 x9))).
Apply pack_c_0_eq2 with
x6,
x7,
λ x8 x9 . x3 ∈ setexp x8 (ap (pack_c x4 x5) 0).
Apply pack_c_0_eq2 with
x4,
x5,
λ x8 x9 . x3 ∈ setexp x6 x8.
The subproof is completed by applying H4.
Let x1 of type ι be given.
Assume H1: x0 x1.
Apply H0 with
x1,
λ x2 . PreContinuousHom 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 → ο) → ο be given.
Apply pack_c_0_eq2 with
x2,
x3,
λ x4 x5 . PreContinuousHom (pack_c x2 x3) (pack_c x2 x3) (lam_id x4).
Apply unknownprop_147946d52b6747e7a3735111f3622ca84b157f241b7b107aab3bab9bb651af48 with
x2,
x2,
x3,
x3,
lam_id x2,
λ x4 x5 : ο . x5.
Apply andI with
lam_id x2 ∈ setexp x2 x2,
∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ x5 ∈ x2) ⟶ x3 x4 ⟶ x3 (λ x5 . and (x5 ∈ x2) (x4 (ap (lam_id x2) x5))) leaving 2 subgoals.
The subproof is completed by applying lam_id_exp_In with x2.
Let x4 of type ι → ο be given.
Assume H2: ∀ x5 . x4 x5 ⟶ x5 ∈ x2.
Assume H3: x3 x4.
Claim L4:
x4 = λ x5 . and (x5 ∈ x2) (x4 (ap (lam_id x2) x5))Apply pred_ext_2 with
x4,
λ x5 . and (x5 ∈ x2) (x4 (ap (lam_id x2) x5)) leaving 2 subgoals.
Let x5 of type ι be given.
Assume H4: x4 x5.
Apply andI with
x5 ∈ x2,
x4 (ap (lam_id ...) ...) leaving 2 subgoals.
Apply L4 with
λ x5 x6 : ι → ο . x3 x5.
The subproof is completed by applying H3.