reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th7:
  for b be bag of X holds
    support (n(#)b) c= support b
proof
  let b be bag of X;
  let x be object such that
A1: x in  support (n(#)b);
  (n(#)b).x <> 0 by A1,PRE_POLY:def 7;
  then x in dom (n(#)b) by FUNCT_1:def 2;
  then (n(#)b).x = n * b.x by VALUED_1:def 5;
  then b.x <>0 by A1,PRE_POLY:def 7;
  hence thesis by PRE_POLY:def 7;
end;
