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 Th8:
  Open_setLatt(T) is lower-bounded & Bottom Open_setLatt(T) = {}
proof
  set OL=Open_setLatt(T);
  reconsider r = {} as Element of OL by PRE_TOPC:1;
A1: now
    let p;
    thus r"/\"p = r /\ p by Def3
      .= r;
    hence p"/\"r=r;
  end;
  thus OL is lower-bounded
  by A1;
  hence thesis by A1,LATTICES:def 16;
end;
