reserve o1,o2 for Ordinal;

theorem Th13:
  for a1,b1,c1 being Element of Bags o1, a2,b2,c2 being Element of
Bags o2 st c1 = b1 -' a1 & c2 = b2 -' a2 holds (b1 +^ b2) -' (a1 +^ a2) = c1 +^
  c2
proof
  let a1,b1,c1 be Element of Bags o1, a2,b2,c2 be Element of Bags o2;
  assume that
A1: c1 = b1 -' a1 and
A2: c2 = b2 -' a2;
  reconsider w = (b1 +^ b2) -' (a1 +^ a2) as Element of Bags(o1+^o2) by
PRE_POLY:def 12;
  for o be Ordinal holds (o in o1 implies w.o = c1.o) & (o in (o1+^o2) \
  o1 implies w.o=c2.(o-^o1))
  proof
    let o be Ordinal;
    hereby
      assume
A3:   o in o1;
      thus w.o = (b1 +^ b2).o -' (a1 +^ a2).o by PRE_POLY:def 6
        .= b1.o -' (a1 +^ a2).o by A3,Def1
        .= b1.o -' a1.o by A3,Def1
        .= c1.o by A1,PRE_POLY:def 6;
    end;
    assume
A4: o in (o1+^o2) \ o1;
    thus w.o = (b1 +^ b2).o -' (a1 +^ a2).o by PRE_POLY:def 6
      .= b2.(o-^o1) -' (a1 +^ a2).o by A4,Def1
      .= b2.(o-^o1) -' a2.(o-^o1) by A4,Def1
      .= c2.(o-^o1) by A2,PRE_POLY:def 6;
  end;
  hence thesis by Def1;
end;
