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

theorem Th44: :: 1.10. Propostion (i), p.104
  for L being complete continuous Scott TopLattice, p be Element of L
  holds { wayabove q where q is Element of L: q << p } is Basis of p
proof
  let L be complete continuous Scott TopLattice, p be Element of L;
  set X = { wayabove q where q is Element of L: q << p };
  X c= bool the carrier of L
  proof
    let e be object;
    assume e in X;
    then ex q being Element of L st e = wayabove q & q << p;
    hence thesis;
  end;
  then reconsider X as Subset-Family of L;
  X is Basis of p
  proof
A1: X is open
    proof
      let e be Subset of L;
      assume e in X;
      then consider q being Element of L such that
A2:   e = wayabove q and q << p;
      wayabove q is open by Th36;
      hence thesis by A2;
    end;
    X is p-quasi_basis
    proof
    for Y being set st Y in X holds p in Y
    proof
      let e be set;
      assume e in X;
      then ex q being Element of L st e = wayabove q & q << p;
      hence thesis;
    end;
    hence p in Intersect X by SETFAM_1:43;
    let S be Subset of L such that
A3: S is open and
A4: p in S;
    consider u being Element of L such that
A5: u << p and
A6: u in S by A3,A4,Th43;
    take V = wayabove u;
    thus V in X by A5;
A7: S is upper by A3,Def4;
A8: wayabove u c= uparrow u by WAYBEL_3:11;
    uparrow u c= S by A6,A7,Th42;
    hence thesis by A8;
  end;
  hence thesis by A1;
  end;
  hence thesis;
end;
