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