Reduce $\overline{K}$ to the set of pairs of natural numbers codifying programs such that each domain and image of the function computed by each of such programs has exactly $1$ element, and the union of the domains and images of both programs has exactly $4$ elements (roughly, the set of pairs of programs whose domains and images have exactly $1$ element, and the union of all such domains and images have exactly $4$ elements) in order to prove that such set is not semi-decidable (not recursively enumerable).