Let x0 of type ι be given.
Apply H0 with
∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x4 x5)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x5 x4)) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0.
Apply H1 with
λ x1 . unpack_b_o x1 (λ x2 . λ x3 : ι → ι → ι . and (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (x3 x4)) (∀ x4 . x4 ∈ x2 ⟶ bij x2 x2 (λ x5 . x3 x5 x4))) ⟶ ∀ x2 : ι → ο . (∀ x3 . ∀ x4 : ι → ι → ι . (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 . x6 ∈ x3 ⟶ x4 x5 x6 ∈ x3) ⟶ (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (λ x6 . x4 x5 x6)) ⟶ (∀ x5 . x5 ∈ x3 ⟶ bij x3 x3 (λ x6 . x4 x6 x5)) ⟶ x2 (pack_b x3 x4)) ⟶ x2 x1.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Assume H2: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ∈ x1.
Apply unknownprop_b2167d2837715e11967d084f35fa508acd069640c7a7c47aa3f84dfcaa1371b5 with
x1,
x2,
λ x3 x4 : ο . x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ x7 x8 x9 ∈ x6) ⟶ (∀ x8 . x8 ∈ x6 ⟶ bij x6 x6 (λ x9 . x7 x8 x9)) ⟶ (∀ x8 . x8 ∈ x6 ⟶ bij x6 x6 (λ x9 . x7 x9 x8)) ⟶ x5 (pack_b x6 x7)) ⟶ x5 (pack_b x1 x2).
Assume H3:
and (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (x2 x3)) (∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (λ x4 . x2 x4 x3)).
Apply H3 with
∀ x3 : ι → ο . (∀ x4 . ∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x4 ⟶ ∀ x7 . x7 ∈ x4 ⟶ x5 x6 x7 ∈ x4) ⟶ (∀ x6 . x6 ∈ x4 ⟶ bij x4 x4 (λ x7 . x5 x6 x7)) ⟶ (∀ x6 . x6 ∈ x4 ⟶ bij x4 x4 (λ x7 . x5 x7 x6)) ⟶ x3 (pack_b x4 x5)) ⟶ x3 (pack_b x1 x2).
Assume H4:
∀ x3 . x3 ∈ x1 ⟶ bij x1 x1 (x2 x3).
Assume H5:
∀ x3 . ... ⟶ bij x1 x1 ....