Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H0:
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) ∈ Sep2 x0 x1 x2.
Apply L1 with
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4).
Let x5 of type ι be given.
Assume H2:
(λ x6 . and (x6 ∈ x0) (∃ x7 . and (x7 ∈ x1 x6) (and (lam 2 (λ x8 . If_i (x8 = 0) x3 x4) = lam 2 (λ x8 . If_i (x8 = 0) x6 x7)) (x2 x6 x7)))) x5.
Apply H2 with
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4).
Assume H3: x5 ∈ x0.
Assume H4:
∃ x6 . and (x6 ∈ x1 x5) (and (lam 2 (λ x7 . If_i (x7 = 0) x3 x4) = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6)).
Apply H4 with
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4).
Let x6 of type ι be given.
Assume H5:
(λ x7 . and (x7 ∈ x1 x5) (and (lam 2 (λ x8 . If_i (x8 = 0) x3 x4) = lam 2 (λ x8 . If_i (x8 = 0) x5 x7)) (x2 x5 x7))) x6.
Apply H5 with
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4).
Assume H6: x6 ∈ x1 x5.
Assume H7:
and (lam 2 (λ x7 . If_i (x7 = 0) x3 x4) = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6).
Apply H7 with
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4).
Assume H8:
lam 2 (λ x7 . If_i (x7 = 0) x3 x4) = lam 2 (λ x7 . If_i (x7 = 0) x5 x6).
Assume H9: x2 x5 x6.
Apply tuple_2_inj with
x3,
x4,
x5,
x6,
and (and (x3 ∈ x0) (x4 ∈ x1 x3)) (x2 x3 x4) leaving 2 subgoals.
The subproof is completed by applying H8.
Assume H10: x3 = x5.
Assume H11: x4 = x6.
Apply H10 with
λ x7 x8 . and (and (x8 ∈ x0) (x4 ∈ x1 x8)) (x2 x8 x4).
Apply H11 with
λ x7 x8 . and (and (x5 ∈ x0) (x8 ∈ ...)) ....