
theorem Th28:
  for R being reflexive antisymmetric non empty RelStr, x being
  Element of R holds (uparrow x) /\ (downarrow x) = {x}
proof
  let R be reflexive antisymmetric non empty RelStr, x be Element of R;
  hereby
    let a be object;
    assume
A1: a in (uparrow x) /\ (downarrow x);
    then reconsider y = a as Element of R;
    y in downarrow x by A1,XBOOLE_0:def 4;
    then
A2: y <= x by WAYBEL_0:17;
    y in uparrow x by A1,XBOOLE_0:def 4;
    then x <= y by WAYBEL_0:18;
    then x = a by A2,ORDERS_2:2;
    hence a in {x} by TARSKI:def 1;
  end;
A3: x <= x;
  then
A4: x in downarrow x by WAYBEL_0:17;
  x in uparrow x by A3,WAYBEL_0:18;
  then x in (uparrow x) /\ (downarrow x) by A4,XBOOLE_0:def 4;
  hence thesis by ZFMISC_1:31;
end;
