reserve o1,o2 for Ordinal;

theorem
  for a be Element of Bags o1,b be Element of Bags o2 st o1 = {} holds a
  +^ b = b
proof
  let a be Element of Bags o1,b be Element of Bags o2;
  assume
A1: o1={};
  then reconsider ab = a +^ b as Element of Bags o2 by ORDINAL2:30;
  now
    let i be object;
    assume
A2: i in o2;
    then reconsider i9=i as Ordinal;
A3: i9-^o1 = i9 by A1,ORDINAL3:56;
    i in (o1+^o2) \ o1 by A1,A2,ORDINAL2:30;
    hence ab.i = b.i by A3,Def1;
  end;
  hence thesis by PBOOLE:3;
end;
