reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th44:
  for n being Ordinal, p, q being bag of n holds p <=' q or q <=' p
proof
  let n be Ordinal, p, q be bag of n;
  assume
A1: not p <=' q;
  then consider i being object such that
A2: i in n and
A3: p.i <> q.i by PBOOLE:3;
A4: not p < q by A1;
  defpred P[set] means p.$1 <> q.$1;
A5: ex i being Ordinal st P[i] by A2,A3;
  consider m being Ordinal such that
A6: P[m] and
A7: for n being Ordinal st P[n] holds m c= n from ORDINAL1:sch 1(A5);
A8: for l being Ordinal st l in m holds q.l = p.l by A7,ORDINAL1:5;
  per cases by A6,XXREAL_0:1;
  suppose
    p.m < q.m;
    hence thesis by A4,A8;
  end;
  suppose
    p.m > q.m;
    then q < p by A8;
    hence thesis;
  end;
end;
