Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Let x2 of type ι be given.
Assume H0: x2 ∈ x0.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Apply prop_ext_2 with
decode_r (encode_r x0 x1) x2 x3,
x1 x2 x3 leaving 2 subgoals.
Assume H2:
lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ Sep2 x0 (λ x4 . x0) x1.
Apply Sep2E'3 with
x0,
λ x4 . x0,
x1,
x2,
x3.
The subproof is completed by applying H2.
Assume H2: x1 x2 x3.
Apply Sep2I with
x0,
λ x4 . x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.