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.
Assume H0:
x3 ∈ Sep2 x0 x1 x2.
Apply SepE with
lam x0 (λ x4 . x1 x4),
λ x4 . x2 (ap x4 0) (ap x4 1),
x3,
∃ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x1 x4) (and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5))) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
x3 ∈ lam x0 (λ x4 . x1 x4).
Assume H2:
x2 (ap x3 0) (ap x3 1).
Apply lamE2 with
x0,
x1,
x3,
∃ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x1 x4) (and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5))) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H3:
(λ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)))) x4.
Apply H3 with
∃ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6))).
Assume H4: x4 ∈ x0.
Assume H5:
∃ x5 . and (x5 ∈ x1 x4) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply H5 with
∃ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6))).
Let x5 of type ι be given.
Assume H6:
(λ x6 . and (x6 ∈ x1 x4) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x4 x6))) x5.
Apply H6 with
∃ x6 . and (x6 ∈ x0) (∃ x7 . and (x7 ∈ x1 x6) (and (x3 = lam 2 (λ x8 . If_i (x8 = 0) x6 x7)) (x2 x6 x7))).
Assume H7: x5 ∈ x1 x4.
Assume H8:
x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Let x6 of type ο be given.
Assume H9:
∀ x7 . and (x7 ∈ x0) (∃ x8 . and (x8 ∈ x1 x7) (and (x3 = lam 2 (λ x9 . If_i (x9 = 0) ... ...)) ...)) ⟶ x6.