reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);

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;
