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

theorem
  for b be bag of X holds
    support (b +*(x,n)) c= {x} \/ support b
proof
  let b be bag of X;
  let y be object;
  assume y in support (b +*(x,n));
  then
A1: (b +*(x,n)).y <>0 by PRE_POLY:def 7;
  per cases;
  suppose y=x;
    then y in {x} by TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose y<>x;
    then b.y<>0 by A1,FUNCT_7:32;
    then y in support b by PRE_POLY:def 7;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
