
theorem
  for L be non empty RelStr for x,y be Element of L st x is_minimal_in (
the carrier of L) \ downarrow y holds (downarrow x) \ {x} = (downarrow x) /\ (
  downarrow y)
proof
  let L be non empty RelStr;
  let x,y be Element of L;
  assume
A1: x is_minimal_in (the carrier of L) \ downarrow y;
  then x in (the carrier of L) \ downarrow y by WAYBEL_4:56;
  then not x in downarrow y by XBOOLE_0:def 5;
  then
A2: not x <= y by WAYBEL_0:17;
  thus (downarrow x) \ {x} c= (downarrow x) /\ (downarrow y)
  proof
    let a be object;
    assume
A3: a in (downarrow 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 downarrow x by A3,XBOOLE_0:def 5;
    then a1 <= x by WAYBEL_0:17;
    then a1 < x by A4,ORDERS_2:def 6;
    then not a1 in (the carrier of L) \ downarrow y by A1,WAYBEL_4:56;
    then a in downarrow y by XBOOLE_0:def 5;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  let a be object;
  assume
A6: a in (downarrow x) /\ (downarrow y);
  then
A7: a in downarrow x by XBOOLE_0:def 4;
  reconsider a1 = a as Element of L by A6;
  a in downarrow y by A6,XBOOLE_0:def 4;
  then a1 <= y by WAYBEL_0:17;
  then not a in {x} by A2,TARSKI:def 1;
  hence thesis by A7,XBOOLE_0:def 5;
end;
