theorem
  for X being Subset of COMPLEX st X is open for z0 be Complex st z0 in
  X holds ex g be Real st {y where y is Complex : |.y-z0.| < g} c= X
proof
  let X be Subset of COMPLEX such that
A1: X is open;
  let z0 be Complex;
  assume z0 in X;
  then consider N be Neighbourhood of z0 such that
A2: N c= X by A1,Th9;
  consider g be Real such that
  0 < g and
A3: {y where y is Complex : |.y-z0.| < g} c= N by Def5;
  take g;
  thus thesis by A2,A3;
end;
