reserve a,b for object, I,J for set;

theorem Lem4:
  for m,n being bag of I holds
  m-'n <> EmptyBag I or m = n or n-'m <> EmptyBag I
  proof
    let m,n be bag of I;
    assume Z0: m-'n = EmptyBag I;
    assume m <> n;
    then consider a such that
A1: a in I & m.a <> n.a by PBOOLE:def 10;
    per cases by A1,XXREAL_0:1;
    suppose m.a > n.a;
      then m.a-n.a > 0 by XREAL_1:50;
      then m.a-'n.a > 0 by XREAL_0:def 2;
      then 0 < (m-'n).a = 0 by A1,Z0,FUNCOP_1:7,PRE_POLY:def 6;
      hence thesis;
    end;
    suppose
A3:   n.a > m.a;
      take a; thus a in I by A1;
      n.a-m.a > 0 by A3,XREAL_1:50;
      then n.a-'m.a > 0 by XREAL_0:def 2;
      then (n-'m).a > 0 by PRE_POLY:def 6;
      hence (n-'m).a <> (EmptyBag I).a by A1,FUNCOP_1:7;
    end;
  end;
