reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem Th45:
  for T being complete continuous Scott TopLattice holds
  the set of all  wayabove x where x is Element of T is Basis of T
proof
  let T be complete continuous Scott TopLattice;
  set B = the set of all  wayabove x where x is Element of T;
A1: B c= the topology of T
  proof
    let e be object;
    assume e in B;
    then consider x being Element of T such that
A2: e = wayabove x;
    wayabove x is open by Th36;
    hence thesis by A2;
  end;
  then reconsider P = B as Subset-Family of T by XBOOLE_1:1;
  for x being Point of T ex B being Basis of x st B c= P
  proof
    let x be Point of T;
    reconsider p = x as Element of T;
    reconsider A =
    { wayabove q where q is Element of T: q << p } as Basis of x by Th44;
    take A;
    let u be object;
    assume u in A;
    then ex q being Element of T st u = wayabove q & q << p;
    hence thesis;
  end;
  hence thesis by A1,YELLOW_8:14;
end;
