
theorem Th9:
  for A being set, b being bag of A st Sum b = 0 holds b = EmptyBag A
proof
  let A be set, b be bag of A;
  set cS = canFS(support b);
  consider f being FinSequence of NAT such that
A1: degree b = Sum f and
A2: f = b*canFS(support b) by Def3;
  assume
A3: degree b = 0;
  now
    assume
A4: support b <> {};
      consider x being object such that
A5:   x in support b by A4,XBOOLE_0:def 1;
      x in rng cS by A5,FUNCT_2:def 3;
      then consider i being Nat such that
A6:   i in dom cS and
A7:   cS.i = x by FINSEQ_2:10;
      reconsider i as Element of NAT by ORDINAL1:def 12;
      f.i = b.(cS.i) by A2,A6,FUNCT_1:13;
      then
A8:   f.i <> 0 by A5,A7,PRE_POLY:def 7;
      support b c= dom b by PRE_POLY:37;
     then
A9:  i in dom f & 0 < f.i
        by A8,A2,A5,A6,A7,FUNCT_1:11;
    for i be Nat st i in dom f holds 0 <= f.i;
    hence contradiction by A3,A1,A9,RVSUM_1:85;
  end;
  hence thesis by PRE_POLY:81;
end;
