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

theorem  :: 1.10. Propostion (iii), p.104
  for T being complete continuous Scott TopLattice, p being Element of T
  holds Int uparrow p = wayabove p
proof
  let T be complete continuous Scott TopLattice, p be Element of T;
  thus Int uparrow p c= wayabove p
  proof
    let y be object;
    assume
A1: y in Int uparrow p;
    then reconsider q = y as Element of T;
    reconsider S = Int uparrow p as Subset of T;
    consider u being Element of T such that
A2: u << q and
A3: u in S by A1,Th43;
    S c= uparrow p by TOPS_1:16;
    then p <= u by A3,WAYBEL_0:18;
    then p << q by A2,WAYBEL_3:2;
    hence thesis;
  end;
  wayabove p c= uparrow p by WAYBEL_3:11;
  hence thesis by Th36,TOPS_1:24;
end;
