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 Th7:
  for A being Subset of GX st A is a_component holds Component_of A=A
proof
  let A be Subset of GX;
  assume
A1: A is a_component;
  then
A2: A is connected;
  then
A3: A c= Component_of A by Th1;
  A <>{}(GX) by A1;
  then Component_of A is connected by A2,Th5;
  hence thesis by A1,A3,CONNSP_1:def 5;
end;
