Let x0 of type ι → ο be given.
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 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.
Apply H3 with
x3 ∈ setexp x6 x4.
Assume H5:
∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ x9 ∈ x6) ⟶ x7 x8 ⟶ x5 (λ x9 . and (x9 ∈ x4) (x8 (ap x3 x9))).
The subproof is completed by applying H4.
Apply unknownprop_cb7abf829499aec888363ff9292dd7680786c42dc92f10fdd88dc16ada048723 with
x0,
λ x1 . ap x1 0,
PreContinuousHom.
The subproof is completed by applying L1.