Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι → ι → ο be given.
Let x5 of type ι → ι → ο be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply pack_r_r_e_0_eq with
pack_r_r_e x0 x2 x4 x6,
x1,
x3,
x5,
x7.
The subproof is completed by applying H0.
Claim L2: x0 = x1
Apply L1 with
λ x8 x9 . x0 = x9.
The subproof is completed by applying pack_r_r_e_0_eq2 with x0, x2, x4, x6.
Apply and4I with
x0 = x1,
∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x2 x8 x9 = x3 x8 x9,
∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x8 x9 = x5 x8 x9,
x6 = x7 leaving 4 subgoals.
The subproof is completed by applying L2.
Let x8 of type ι be given.
Assume H3: x8 ∈ x0.
Let x9 of type ι be given.
Assume H4: x9 ∈ x0.
Apply pack_r_r_e_1_eq2 with
x0,
x2,
x4,
x6,
x8,
x9,
λ x10 x11 : ο . x11 = x3 x8 x9 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Claim L5: x8 ∈ x1
Apply L2 with
λ x10 x11 . x8 ∈ x10.
The subproof is completed by applying H3.
Claim L6: x9 ∈ x1
Apply L2 with
λ x10 x11 . x9 ∈ x10.
The subproof is completed by applying H4.
Apply H0 with
λ x10 x11 . decode_r (ap x11 1) x8 x9 = x3 x8 x9.
Let x10 of type ο → ο → ο be given.
Apply pack_r_r_e_1_eq2 with
x1,
x3,
x5,
x7,
x8,
x9,
λ x11 x12 : ο . x10 x12 x11 leaving 2 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.
Let x8 of type ι be given.
Assume H3: x8 ∈ x0.
Let x9 of type ι be given.
Assume H4: x9 ∈ x0.
Apply pack_r_r_e_2_eq2 with
x0,
x2,
x4,
x6,
x8,
x9,
λ x10 x11 : ο . x11 = x5 x8 x9 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Claim L5: x8 ∈ x1
Apply L2 with
λ x10 x11 . x8 ∈ x10.
The subproof is completed by applying H3.
Claim L6: x9 ∈ x1
Apply L2 with
λ x10 x11 . x9 ∈ x10.
The subproof is completed by applying H4.
Apply H0 with
λ x10 x11 . decode_r (ap x11 2) x8 x9 = x5 x8 x9.
Let x10 of type ο → ο → ο be given.
Apply pack_r_r_e_2_eq2 with
x1,
x3,
x5,
x7,
x8,
x9,
λ x11 x12 : ο . x10 x12 x11 leaving 2 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.
Apply pack_r_r_e_3_eq2 with
x0,
x2,
x4,
x6,
λ x8 x9 . x9 = x7.
Apply H0 with
λ x8 x9 . ap x9 3 = x7.
Let x8 of type ι → ι → ο be given.
The subproof is completed by applying pack_r_r_e_3_eq2 with x1, x3, x5, x7, λ x9 x10 . x8 x10 x9.