Claim L0:
∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ο . (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ (λ x5 x6 x7 . λ x8 : ι → ι → ο . pack_r {x9 ∈ x7|ap x5 x9 = ap x6 x9} x8) x0 x1 x2 x4 = (λ x5 x6 x7 . λ x8 : ι → ι → ο . pack_r {x9 ∈ x7|ap x5 x9 = ap x6 x9} x8) x0 x1 x2 x3Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι → ι → ο be given.
Assume H0:
∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ iff (x3 x5 x6) (x4 x5 x6).
Apply pack_r_ext with
{x5 ∈ x2|ap x0 x5 = ap x1 x5},
x4,
x3.
Let x5 of type ι be given.
Assume H1:
x5 ∈ {x6 ∈ x2|ap x0 x6 = ap x1 x6}.
Let x6 of type ι be given.
Assume H2:
x6 ∈ {x7 ∈ x2|ap x0 x7 = ap x1 x7}.
Apply H0 with
x5,
x6,
iff (x4 x5 x6) (x3 x5 x6) leaving 3 subgoals.
Apply SepE1 with
x2,
λ x7 . ap x0 x7 = ap x1 x7,
x5.
The subproof is completed by applying H1.
Apply SepE1 with
x2,
λ x7 . ap x0 x7 = ap x1 x7,
x6.
The subproof is completed by applying H2.
Assume H3: x3 x5 x6 ⟶ x4 x5 x6.
Assume H4: x4 x5 x6 ⟶ x3 x5 x6.
Apply iffI with
x4 x5 x6,
x3 x5 x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Apply unpack_r_i_eq with
(λ x5 x6 x7 . λ x8 : ι → ι → ο . pack_r {x9 ∈ x7|ap x5 x9 = ap x6 x9} x8) x3 x4,
x0,
x1.
The subproof is completed by applying L0 with x3, x4, x0, x1.