theorem Th14:
  for phi being Ordinal-Sequence st phi is increasing holds C+^phi
  is increasing
proof
  let phi be Ordinal-Sequence such that
A1: phi is increasing;
  let A,B;
  set xi = C+^phi;
  assume that
A2: A in B and
A3: B in dom xi;
  reconsider A9 = phi.A, B9 = phi.B as Ordinal;
A4: dom xi = dom phi by ORDINAL3:def 1;
  then
A5: xi.B = C+^B9 by A3,ORDINAL3:def 1;
  A in dom xi by A2,A3,ORDINAL1:10;
  then
A6: xi.A = C+^A9 by A4,ORDINAL3:def 1;
  A9 in B9 by A1,A2,A3,A4;
  hence thesis by A6,A5,ORDINAL2:32;
end;
