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

theorem Th16:
  for b be bag of X holds
    support (b +*(x,0)) = support b\{x}
proof
  let b be bag of X;
A1:dom (b +*(x,0)) = X = dom b by PARTFUN1:def 2;
  thus support (b +*(x,0)) c= support b\{x}
  proof
    let y be object;
    assume
A2: y in support (b +*(x,0)); then
A3:(b +*(x,0)).y<>0 by PRE_POLY:def 7;
A4: y<>x by A2,A3,A1,FUNCT_7:31;
    then 0 <> b.y by A3,FUNCT_7:32;
    then y in support b by PRE_POLY:def 7;
    hence thesis by A4,ZFMISC_1:56;
  end;
  let y be object;
  assume y in support b\{x};
  then
A5: y in support b c= dom b & x <>y by ZFMISC_1:56,PRE_POLY:37;
  then 0 <> b.y & y in dom b by PRE_POLY:def 7;
  then (b +*(x,0)).y<>0 by A5,FUNCT_7:32;
  hence thesis by PRE_POLY:def 7;
end;
