theorem
  A` is Fo_closed implies A is Fo_open
proof
  assume A` is Fo_closed;
  then
A1: (A`) = (A`)^Fob;
  (A`)^Fob = (((A`)`)^Foi)` by Th38
    .= (A^Foi)`;
  then A = (A^Foi)`` by A1
    .= A^Foi;
  hence thesis;
end;
