
theorem
  for N being non empty reflexive RelStr, A, J being Subset of N st A c=
  J holds A^0 c= J^0
proof
  let N be non empty reflexive RelStr, A, J be Subset of N such that
A1: A c= J;
  let a be object;
  assume a in A^0;
  then consider u being Element of N such that
A2: a = u and
A3: for D being non empty directed Subset of N st u <= sup D holds A meets D;
  for D being non empty directed Subset of N st u <= sup D holds J meets D
  by A1,A3,XBOOLE_1:63;
  hence thesis by A2;
end;
