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

theorem Th41: :: 1.9 Remark, p.104
  for T being complete Scott TopLattice,
  S being upper Subset of T st S is Open holds S is open
proof
  let T be complete Scott TopLattice, S be upper Subset of T such that
A1: for x be Element of T st x in S ex y be Element of T st y in S & y << x;
  S is inaccessible
  proof
    let D be non empty directed Subset of T;
    assume sup D in S;
    then consider y being Element of T such that
A2: y in S and
A3: y << sup D by A1;
    consider d being Element of T such that
A4: d in D and
A5: y <= d by A3;
    d in S by A2,A5,WAYBEL_0:def 20;
    hence thesis by A4,XBOOLE_0:3;
  end;
  hence thesis by Def4;
end;
