Let x0 of type ι be given.
Apply H0 with
∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ο . ∀ x4 . x4 ∈ x2 ⟶ x3 x4 ⟶ x1 (pack_p x2 x3)) ⟶ x1 x0.
Apply H1 with
λ x1 . unpack_p_o x1 (λ x2 . λ x3 : ι → ο . ∃ x4 . and (x4 ∈ x2) (x3 x4)) ⟶ ∀ x2 : ι → ο . (∀ x3 . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x3 ⟶ x4 x5 ⟶ x2 (pack_p x3 x4)) ⟶ x2 x1.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Apply unpack_p_o_eq with
λ x3 . λ x4 : ι → ο . ∃ x5 . and (x5 ∈ x3) (x4 x5),
x1,
x2,
λ x3 x4 : ο . x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . ∀ x7 : ι → ο . ∀ x8 . x8 ∈ x6 ⟶ x7 x8 ⟶ x5 (pack_p x6 x7)) ⟶ x5 (pack_p x1 x2) leaving 2 subgoals.
The subproof is completed by applying unknownprop_b2592fa24ced3d84e007e876699c34520860460028b2dc9144c228a16f98ce34 with x1, x2.
Assume H2:
∃ x3 . and (x3 ∈ x1) (x2 x3).
Apply H2 with
∀ x3 : ι → ο . (∀ x4 . ∀ x5 : ι → ο . ∀ x6 . x6 ∈ x4 ⟶ x5 x6 ⟶ x3 (pack_p x4 x5)) ⟶ x3 (pack_p x1 x2).
Let x3 of type ι be given.
Assume H3:
(λ x4 . and (x4 ∈ x1) (x2 x4)) x3.
Apply H3 with
∀ x4 : ι → ο . (∀ x5 . ∀ x6 : ι → ο . ∀ x7 . x7 ∈ x5 ⟶ x6 x7 ⟶ x4 (pack_p x5 x6)) ⟶ x4 (pack_p x1 x2).
Assume H4: x3 ∈ x1.
Assume H5: x2 x3.
Let x4 of type ι → ο be given.
Assume H6:
∀ x5 . ∀ x6 : ι → ο . ∀ x7 . x7 ∈ x5 ⟶ x6 x7 ⟶ x4 (pack_p x5 x6).
Apply H6 with
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.