theorem
  (for A st A in dom fi holds fi.A = C+^A) implies fi is increasing
proof
  assume
A1: for A st A in dom fi holds fi.A = C+^A;
  let A,B;
  assume that
A2: A in B and
A3: B in dom fi;
A4: fi.B = C+^B by A1,A3;
  fi.A = C+^A by A1,A2,A3,ORDINAL1:10;
  hence thesis by A2,A4,ORDINAL2:32;
end;
