reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem
  for V being Subset of GX st V is connected holds Component_of V <>{}
proof
  let V be Subset of GX such that
A1: V is connected;
  per cases;
  suppose
    V = {};
    then V = {}GX;
    hence thesis by Th3;
  end;
  suppose
    V <>{};
    hence thesis by A1,Th1,XBOOLE_1:3;
  end;
end;
