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 Th40:
  for n being Ordinal, p, q, r being bag of n st p < q & q < r holds p < r
proof
  let n be Ordinal, p, q, r be bag of n;
  assume that
A1: p < q and
A2: q < r;
  consider k being Ordinal such that
A3: p.k < q.k and
A4: for l being Ordinal st l in k holds p.l = q.l by A1;
  consider m being Ordinal such that
A5: q.m < r.m and
A6: for l being Ordinal st l in m holds q.l = r.l by A2;
  take n = k /\ m;
A7: n c= m & n <> m iff n c< m;
A8: n c= k & n <> k iff n c< k;
  now
    per cases by ORDINAL1:14;
    suppose
      k in m;
      hence p.n < r.n by A3,A6,A8,A7,ORDINAL1:11,ORDINAL3:13,XBOOLE_1:17;
    end;
    suppose
      m in k;
      hence p.n < r.n by A4,A5,A8,A7,ORDINAL1:11,ORDINAL3:13,XBOOLE_1:17;
    end;
    suppose
      m = k;
      hence p.n < r.n by A3,A5,XXREAL_0:2;
    end;
  end;
  hence p.n < r.n;
  let l be Ordinal;
A9: n c= m by XBOOLE_1:17;
  assume
A10: l in n;
  n c= k by XBOOLE_1:17;
  hence p.l = q.l by A4,A10
    .= r.l by A6,A9,A10;
end;
