Let x0 of type ι be given.
Apply H0 with
∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 = x3 x5 ⟶ x4 = x5) ⟶ x1 (pack_u x2 x3)) ⟶ x1 x0.
Apply H1 with
λ x1 . unpack_u_o x1 (λ x2 . inj x2 x2) ⟶ ∀ x2 : ι → ο . (∀ x3 . ∀ x4 : ι → ι . (∀ x5 . x5 ∈ x3 ⟶ x4 x5 ∈ x3) ⟶ (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 . x6 ∈ x3 ⟶ x4 x5 = x4 x6 ⟶ x5 = x6) ⟶ x2 (pack_u x3 x4)) ⟶ x2 x1.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Apply unknownprop_a16c1e4668d0beb05131d186b11d5f445a514d4e4db3d676d988ae675bef6e8e with
x1,
x2,
λ x3 x4 : ο . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 ∈ x1) ⟶ x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x6 ⟶ x7 x8 ∈ x6) ⟶ (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ x7 x8 = x7 x9 ⟶ x8 = x9) ⟶ x5 (pack_u x6 x7)) ⟶ x5 (pack_u x1 x2).
Assume H2: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1.
Apply H3 with
∀ x3 : ι → ο . (∀ x4 . ∀ x5 : ι → ι . (∀ x6 . x6 ∈ x4 ⟶ x5 x6 ∈ x4) ⟶ (∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7) ⟶ x3 (pack_u x4 x5)) ⟶ x3 (pack_u x1 x2).
Assume H4: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1.
Assume H5: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Let x3 of type ι → ο be given.
Assume H6:
∀ x4 . ∀ x5 : ι → ι . (∀ x6 . x6 ∈ x4 ⟶ x5 x6 ∈ x4) ⟶ (∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ x5 x6 = x5 x7 ⟶ x6 = x7) ⟶ x3 (pack_u x4 x5).
Apply H6 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.