
theorem Th3:
  for L be non empty RelStr for x,y be Element of L st x
is_maximal_in (the carrier of L) \ uparrow y holds (uparrow x) \ {x} = (uparrow
  x) /\ (uparrow y)
proof
  let L be non empty RelStr;
  let x,y be Element of L;
  assume
A1: x is_maximal_in (the carrier of L) \ uparrow y;
  then x in (the carrier of L) \ uparrow y by WAYBEL_4:55;
  then not x in uparrow y by XBOOLE_0:def 5;
  then
A2: not y <= x by WAYBEL_0:18;
  thus (uparrow x) \ {x} c= (uparrow x) /\ (uparrow y)
  proof
    let a be object;
    assume
A3: a in (uparrow x) \ {x};
    then reconsider a1 = a as Element of L;
    not a in {x} by A3,XBOOLE_0:def 5;
    then
A4: a1 <> x by TARSKI:def 1;
A5: a in uparrow x by A3,XBOOLE_0:def 5;
    then x <= a1 by WAYBEL_0:18;
    then x < a1 by A4,ORDERS_2:def 6;
    then not a1 in (the carrier of L) \ uparrow y by A1,WAYBEL_4:55;
    then a in uparrow y by XBOOLE_0:def 5;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  let a be object;
  assume
A6: a in (uparrow x) /\ (uparrow y);
  then
A7: a in uparrow x by XBOOLE_0:def 4;
  reconsider a1 = a as Element of L by A6;
  a in uparrow y by A6,XBOOLE_0:def 4;
  then y <= a1 by WAYBEL_0:18;
  then not a in {x} by A2,TARSKI:def 1;
  hence thesis by A7,XBOOLE_0:def 5;
end;
