set Z = 0_ n,S;

now

given x being set such that A1: x in Support (0_ n,S) ; :: thesis: contradiction
reconsider x = x as Element of Bags n by A1;
(0_ n,S) . x = 0. S by FUNCOP_1:13;
hence contradiction by A1, Def9; :: thesis: verum

end;

then Support (0_ n,S) = {} by XBOOLE_0:def 1;
hence 0_ n,S is finite-Support by Def10; :: thesis: verum