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 Th48:
  for n being Ordinal, d, b being bag of n st d divides b holds d <=' b
proof
  let n be Ordinal, d, b be bag of n;
  assume that
A1: d divides b and
A2: not d < b;
  now
    defpred P[set] means d.$1 < b.$1;
    let p be object;
    assume p in n;
    then reconsider p9 = p as Ordinal;
    assume
A3: d.p <> b.p;
    d.p9 <= b.p9 by A1;
    then d.p9 < b.p9 by A3,XXREAL_0:1;
    then
A4: ex p being Ordinal st P[p];
    consider k being Ordinal such that
A5: P[k] and
A6: for m being Ordinal st P[m] holds k c= m from ORDINAL1:sch 1(A4);
    now
      let l be Ordinal;
      assume l in k;
      then
A7:   b.l <= d.l by A6,ORDINAL1:5;
      d.l <= b.l by A1;
      hence d.l = b.l by A7,XXREAL_0:1;
    end;
    hence contradiction by A2,A5;
  end;
  hence thesis;
end;
