
theorem Th26:
  for N being Scott TopLattice, X being upper Subset of N holds Int X c= X^0
proof
  let N be Scott TopLattice, X be upper Subset of N;
  let x be object;
  assume
A1: x in Int X;
  then reconsider y = x as Element of N;
  now
A2: Int X is upper inaccessible by WAYBEL11:def 4;
    let D be non empty directed Subset of N;
    assume y <= sup D;
    then sup D in Int X by A2,A1;
    then D meets Int X by A2;
    hence X meets D by TOPS_1:16,XBOOLE_1:63;
  end;
  hence thesis;
end;
