Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ {proj1 x3|x3 ∈ x0,∃ x4 . x3 = setsum x1 x4}.
Apply ReplSepE_impred with
x0,
λ x3 . ∃ x4 . x3 = setsum x1 x4,
proj1,
x2,
setsum x1 x2 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Assume H2:
∃ x4 . x3 = setsum x1 x4.
Assume H3:
x2 = proj1 x3.
Apply H2 with
setsum x1 x2 ∈ x0.
Let x4 of type ι be given.
Claim L5: x2 = x4
Apply H3 with
λ x5 x6 . x6 = x4.
Apply H4 with
λ x5 x6 . proj1 x6 = x4.
The subproof is completed by applying proj1_pair_eq with x1, x4.
Apply L5 with
λ x5 x6 . x3 = setsum x1 x6.
The subproof is completed by applying H4.
Apply L6 with
λ x5 x6 . x5 ∈ x0.
The subproof is completed by applying H1.