
theorem    ::Remark 1.4 (v)
  for T being Scott up-complete non empty reflexive transitive antisymmetric
  TopRelStr, S being Subset of T holds S is open iff S is upper property(S)
proof
  let T be Scott up-complete non empty reflexive transitive antisymmetric
  TopRelStr, S be Subset of T;
  hereby
    assume
A1: S is open;
    hence
    S is upper;
    thus S is property(S)
    proof
      let D be non empty directed Subset of T;
      assume sup D in S;
      then S meets D by A1,WAYBEL11:def 1;
      then consider y being object such that
A2:   y in S and
A3:   y in D by XBOOLE_0:3;
      reconsider y as Element of T by A2;
      take y;
      thus thesis by A1,A2,A3,WAYBEL_0:def 20;
    end;
  end;
  assume that
A4: S is upper and
A5: S is property(S);
  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
A6: y in D and
A7: for x being Element of T st x in D & x >= y holds x in S by A5;
    y >= y by YELLOW_0:def 1;
    then y in S by A6,A7;
    hence thesis by A6,XBOOLE_0:3;
  end;
  hence thesis by A4;
end;
