reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;
reserve X,Y for non empty TopSpace;

theorem
  for Y being anti-discrete non empty TopSpace, f being Function of X,Y
  holds f is continuous
proof
  let Y be anti-discrete non empty TopSpace, f be Function of X,Y;
  now
    let B be Subset of Y;
    assume
A1: B is closed;
    now
      per cases by A1,TDLAT_3:19;
      suppose
        B = {};
        then f" B = {}X;
        hence f" B is closed;
      end;
      suppose
        B = the carrier of Y;
        then B = [#]Y;
        then f" B = [#]X by TOPS_2:41;
        hence f" B is closed;
      end;
    end;
    hence f" B is closed;
  end;
  hence thesis;
end;
