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 → ι → ι be given.
Let x4 of type ι be given.
Assume H0: x4 ∈ x0.
Let x5 of type ι be given.
Assume H1: x5 ∈ x1 x4.
Assume H2: x2 x4 x5.
Apply UnionI with
{{x3 x6 x7|x7 ∈ x1 x6,x2 x6 x7}|x6 ∈ x0},
x3 x4 x5,
{x3 x4 x6|x6 ∈ x1 x4,x2 x4 x6} leaving 2 subgoals.
Apply ReplSepI with
x1 x4,
x2 x4,
x3 x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply ReplI with
x0,
λ x6 . {x3 x6 x7|x7 ∈ x1 x6,x2 x6 x7},
x4.
The subproof is completed by applying H0.