theorem Th6:
  f is_differentiable_on Y implies Y is open
proof
  assume A1: f is_differentiable_on Y;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
A2: Y c= dom g by A1;
    now
      let x;
      assume x in Y;
      then f|Y is_differentiable_in x by A1;
      hence g|Y is_differentiable_in x;
    end;
    then g is_differentiable_on Y by A2,NDIFF_3:def 5;
    hence Y is open by NDIFF_3:11;
end;
