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 Th8:
  for A being Subset of GX holds A is a_component iff ex V
  being Subset of GX st V is connected & V <> {} & A = Component_of V
proof
  let A be Subset of GX;
A1: now
    assume
A2: A is a_component;
    take V = A;
    thus V is connected & V<>{} & A = Component_of V by A2,Th7;
  end;
  now
    given V being Subset of GX such that
A3: V is connected & V<>{} & A = Component_of V;
    A is connected & for B being Subset of GX st B is connected holds A c=
    B implies A = B by A3,Th5,Th6;
    hence A is a_component by CONNSP_1:def 5;
  end;
  hence thesis by A1;
end;
