Let x0 of type ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι be given.
Assume H0: ∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5.
Apply pack_b_b_b_e_0_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 1)) (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 2)) (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 3)) (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 4) = x0 x1 x2 x3 x4 x5.
Apply pack_b_b_b_e_4_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 1)) (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 2)) (decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 3)) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 1),
decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 2),
decode_b (ap (pack_b_b_b_e x1 x2 x3 x4 x5) 3) leaving 3 subgoals.
The subproof is completed by applying pack_b_b_b_e_1_eq2 with x1, x2, x3, x4, x5.
The subproof is completed by applying pack_b_b_b_e_2_eq2 with x1, x2, x3, x4, x5.
The subproof is completed by applying pack_b_b_b_e_3_eq2 with x1, x2, x3, x4, x5.