Monoid of Multisets and Subsets

We adopt the following convention: x, y are arbitrary, X, Y , Z are sets, and n is a natural number. We now define two new constructions. Let D be a nonempty set, and let d be an element of D. Then {d} is a non-empty subset of D. Let D be a non-empty set, and let X1, X2 be subsets of D. Then X1 ∪X2 is a subset of D. Let D be a non-empty set, and let X1 be a subset of D, and let X2 be a non-empty subset of D. Then X1 ∪X2 is a non-empty subset of D. Let D1, D2, D be non-empty sets. A binary function from D1, D2 into D is a function from [:D1, D2 :] into D. Let f be a function, and let x1, x2, y be arbitrary. The functor f(x1, x2)(y) is defined by: (Def.1) f(x1, x2)(y) = f(〈x1, x2〉)(y). The following proposition is true