Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Apply H0 with
λ x3 . x3 = pack_b_b x0 x1 x2 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 ∈ x0 leaving 2 subgoals.
Let x3 of type ι be given.
Let x4 of type ι → ι → ι be given.
Assume H1: ∀ x5 . x5 ∈ x3 ⟶ ∀ x6 . x6 ∈ x3 ⟶ x4 x5 x6 ∈ x3.
Let x5 of type ι → ι → ι be given.
Assume H2: ∀ x6 . x6 ∈ x3 ⟶ ∀ x7 . x7 ∈ x3 ⟶ x5 x6 x7 ∈ x3.
Apply pack_b_b_inj with
x3,
x0,
x4,
x1,
x5,
x2,
∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H4:
and (x3 = x0) (∀ x6 . x6 ∈ x3 ⟶ ∀ x7 . x7 ∈ x3 ⟶ x4 x6 x7 = x1 x6 x7).
Apply H4 with
(∀ x6 . x6 ∈ x3 ⟶ ∀ x7 . x7 ∈ x3 ⟶ x5 x6 x7 = x2 x6 x7) ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ∈ x0.
Assume H5: x3 = x0.
Assume H6: ∀ x6 . x6 ∈ x3 ⟶ ∀ x7 . x7 ∈ x3 ⟶ x4 x6 x7 = x1 x6 x7.
Assume H7: ∀ x6 . x6 ∈ x3 ⟶ ∀ x7 . x7 ∈ x3 ⟶ x5 x6 x7 = x2 x6 x7.
Apply H5 with
λ x6 x7 . ∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ x2 x8 x9 ∈ x6.
Let x6 of type ι be given.
Assume H8: x6 ∈ x3.
Let x7 of type ι be given.
Assume H9: x7 ∈ x3.
Apply H7 with
x6,
x7,
λ x8 x9 . x8 ∈ x3 leaving 3 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply H2 with
x6,
x7 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Let x3 of type ι → ι → ο be given.
The subproof is completed by applying H1.