reserve o1,o2 for Ordinal;

theorem Th5:
  for b1 be Element of Bags o1,b2 be Element of Bags o2 holds b1+^
  b2 = EmptyBag(o1+^o2) iff b1 = EmptyBag o1 & b2 = EmptyBag o2
proof
  let b1 be Element of Bags o1,b2 be Element of Bags o2;
  hereby
    assume
A1: b1+^b2 = EmptyBag (o1+^o2);
A2: for z be object st z in dom b1 holds b1.z = 0
    proof
      let z be object;
A3:   dom b1 = o1 by PARTFUN1:def 2;
      assume
A4:   z in dom b1;
      then reconsider o=z as Ordinal by A3;
      b1.o = (b1+^b2).o by A4,A3,Def1
        .= 0 by A1,PBOOLE:5;
      hence thesis;
    end;
    dom b1 = o1 by PARTFUN1:def 2;
    then b1 = o1 --> 0 by A2,FUNCOP_1:11;
    hence b1=EmptyBag o1 by PBOOLE:def 3;
A5: for z be object st z in dom b2 holds b2.z = 0
    proof
      let z be object;
A6:   dom b2 = o2 by PARTFUN1:def 2;
      assume
A7:   z in dom b2;
      then reconsider o=z as Ordinal by A6;
      o1 c= o1+^o by ORDINAL3:24;
      then
A8:   not o1+^o in o1 by ORDINAL1:5;
      o1+^o in o1+^o2 by A7,A6,ORDINAL2:32;
      then o1+^o in (o1+^o2) \ o1 by A8,XBOOLE_0:def 5;
      then (b1+^b2).(o1+^o) = b2.(o1+^o-^o1) by Def1;
      then b2.(o1+^o-^o1) = 0 by A1,PBOOLE:5;
      hence thesis by ORDINAL3:52;
    end;
    dom b2 = o2 by PARTFUN1:def 2;
    then b2 = o2 --> 0 by A5,FUNCOP_1:11;
    hence b2=EmptyBag o2 by PBOOLE:def 3;
  end;
  thus b1 = EmptyBag o1 & b2 = EmptyBag o2 implies b1+^b2 = EmptyBag(o1+^o2)
  proof
    assume that
A9: b1 = EmptyBag o1 and
A10: b2 = EmptyBag o2;
A11: for z be object st z in dom (b1+^b2) holds (b1+^b2).z = 0
    proof
      let z be object;
A12:  dom (b1+^b2) = o1+^o2 by PARTFUN1:def 2;
      assume
A13:  z in dom (b1+^b2);
      then reconsider o=z as Ordinal by A12;
A14:  o in (o1+^o2) \ o1 implies b2.(o-^o1) = 0 & (b1+^b2).o=b2.(o-^o1)
      by A10,Def1,PBOOLE:5;
      o in o1 implies b1.o = 0 & (b1+^b2).o=b1.o by A9,Def1,PBOOLE:5;
      hence thesis by A13,A12,A14,XBOOLE_0:def 5;
    end;
    dom (b1+^b2) = o1+^o2 by PARTFUN1:def 2;
    then b1+^b2 = (o1+^o2) --> 0 by A11,FUNCOP_1:11;
    hence thesis by PBOOLE:def 3;
  end;
end;
