reserve o1,o2 for Ordinal;

theorem Th2:
  for n being Ordinal, a,b being bag of n st a < b ex o being
  Ordinal st o in n & a.o < b.o & for l being Ordinal st l in o holds a.l = b.l
proof
  let n be Ordinal, a,b be bag of n;
  assume a < b;
  then consider o being Ordinal such that
A1: a.o < b.o and
A2: for l being Ordinal st l in o holds a.l = b.l by PRE_POLY:def 9;
  take o;
  now
    assume
A3: not o in n;
    then
A4: not o in dom b by PARTFUN1:def 2;
    n = dom a by PARTFUN1:def 2;
    then a.o = 0 by A3,FUNCT_1:def 2;
    hence contradiction by A1,A4,FUNCT_1:def 2;
  end;
  hence o in n;
  thus a.o < b.o by A1;
  thus thesis by A2;
end;
