reserve o1,o2 for Ordinal;

theorem Th9:
  for b1,c1 being Element of Bags o1, b2,c2 being Element of Bags
  o2 st b1 divides c1 & b2 divides c2 holds b1 +^ b2 divides c1 +^ c2
proof
  let b1,c1 be Element of Bags o1, b2,c2 be Element of Bags o2;
  assume that
A1: b1 divides c1 and
A2: b2 divides c2;
  now
    reconsider b19 = b1,c19 = c1 as bag of o1;
    let k be object;
A3: b19.k <= c19.k by A1,PRE_POLY:def 11;
    assume
A4: k in o1+^o2;
    per cases by A4,XBOOLE_0:def 5;
    suppose
A5:   k in o1;
      then reconsider k9=k as Ordinal;
      (b1 +^ b2).k9 = b1.k9 by A5,Def1;
      hence (b1 +^ b2).k <= (c1 +^ c2).k by A3,A5,Def1;
    end;
    suppose
A6:   k in (o1+^ o2)\o1;
      then reconsider k9=k as Ordinal;
      (b1 +^ b2).k9 = b2.(k9-^o1) & (c1 +^ c2).k9 = c2.(k9-^o1) by A6,Def1;
      hence (b1 +^ b2).k <= (c1 +^ c2).k by A2,PRE_POLY:def 11;
    end;
  end;
  hence thesis by PRE_POLY:46;
end;
