theorem
  f is_differentiable_on Y implies Y is open
proof
  assume
A1: f is_differentiable_on Y;
  then
A2: Y c= dom f;
  now
    let x0 be Complex;
    assume x0 in Y;
    then
A3: x0 in dom (f|Y) by A2,RELAT_1:57;
    f|Y is differentiable by A1;
    then f|Y is_differentiable_in x0 by A3;
    then consider N being Neighbourhood of x0 such that
A4: N c= dom(f|Y) and
    ex L,R st for x st x in N holds (f|Y)/.x-(f|Y)/.x0 = L/.(x-x0)+R/.(x -
    x0);
    take N;
    dom(f|Y) = dom f/\Y by RELAT_1:61;
    then dom(f|Y) c= Y by XBOOLE_1:17;
    hence N c= Y by A4;
  end;
  hence thesis by Th11;
end;
