
theorem bag6:
for Z being non empty set
for B1,B2 being bag of Z
st B2 divides B1 & B1 -' B2 is zero holds B2 = B1
proof
let Z be non empty set, B1,B2 be bag of Z;
assume A: B2 divides B1 & B1 -' B2 is zero;
now assume B2 <> B1; then
  consider i being object such that
  C: i in Z & B2.i <> B1.i by PBOOLE:3;
  reconsider n = B1.i - B2.i as Element of NAT
      by INT_1:5,A,PRE_POLY:def 11;
  n <> 0 by C; then
  B1.i -' B2.i > 0 by XREAL_0:def 2; then
  (B1-'B2).i > 0 by PRE_POLY:def 6; then
  i in support(B1-'B2) by PRE_POLY:def 7;
  hence contradiction by A,RING_5:24;
  end;
hence thesis;
end;
