
theorem Th25:
  for N being non empty reflexive RelStr, x being Element of N
  holds (uparrow x)^0 = wayabove x
proof
  let N be non empty reflexive RelStr, x be Element of N;
  thus (uparrow x)^0 c= wayabove x
  proof
    let a be object;
    assume a in (uparrow x)^0;
    then consider u being Element of N such that
A1: a = u and
A2: for D being non empty directed Subset of N st u <= sup D holds (
    uparrow x) meets D;
    u >> x
    proof
      let D be non empty directed Subset of N;
      assume u <= sup D;
      then (uparrow x) meets D by A2;
      then consider d being object such that
A3:   d in (uparrow x) /\ D by XBOOLE_0:4;
      reconsider d as Element of N by A3;
      take d;
      thus d in D by A3,XBOOLE_0:def 4;
      d in uparrow x by A3,XBOOLE_0:def 4;
      hence thesis by WAYBEL_0:18;
    end;
    hence thesis by A1;
  end;
  let a be object;
  assume
A4: a in wayabove x;
  then reconsider b = a as Element of N;
  now
A5: b >> x by A4,WAYBEL_3:8;
    let D be non empty directed Subset of N;
    assume b <= sup D;
    then consider d being Element of N such that
A6: d in D and
A7: x <= d by A5;
    d in uparrow x by A7,WAYBEL_0:18;
    hence (uparrow x) meets D by A6,XBOOLE_0:3;
  end;
  hence thesis;
end;
