theorem Th9:
  Top Open_setLatt(T) = the carrier of T
proof
  set OL=Open_setLatt(T);
  reconsider B = the carrier of T as Element of OL by PRE_TOPC:def 1;
  now
    let p be Element of OL;
    reconsider p9=p as Element of Topology_of T;
    thus B"\/"p = (the carrier of T) \/ p9 by Def2
      .= B by XBOOLE_1:12;
    hence p"\/"B = B;
  end;
  hence thesis by LATTICES:def 17;
end;
