Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ x1 x2 x2.
Assume H1: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2.
Assume H2: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4.
Apply andI with
struct_r (pack_r x0 x1),
unpack_r_o (pack_r x0 x1) (λ x2 . λ x3 : ι → ι → ο . and (and (∀ x4 . x4 ∈ x2 ⟶ x3 x4 x4) (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x4)) (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6)) leaving 2 subgoals.
The subproof is completed by applying pack_struct_r_I with x0, x1.
Apply unknownprop_f2fd2dc1110d448d4f07550eb7e519d4a54ff9bc8af712fc75384622bff0fff8 with
x0,
x1,
λ x2 x3 : ο . x3.
Apply and3I with
∀ x2 . x2 ∈ x0 ⟶ x1 x2 x2,
∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2,
∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.