
theorem
  for L being antisymmetric reflexive with_infima RelStr, y being
  Element of L holds (y"/\").:(uparrow y) = {y}
proof
  let L be antisymmetric reflexive with_infima RelStr, y be Element of L;
  thus (y"/\").:(uparrow y) c= {y}
  proof
    let q be object;
    assume q in (y"/\").:(uparrow y);
    then consider a being object such that
    a in dom (y"/\") and
A1: a in uparrow y and
A2: q = (y"/\").a by FUNCT_1:def 6;
    reconsider a as Element of L by A1;
A3: y <= a by A1,WAYBEL_0:18;
    q = y "/\" a by A2,WAYBEL_1:def 18
      .= y by A3,YELLOW_0:25;
    hence thesis by TARSKI:def 1;
  end;
  let q be object;
  assume q in {y};
  then
A4: q = y by TARSKI:def 1;
  y <= y;
  then
A5: dom (y"/\") = the carrier of L & y in uparrow y by FUNCT_2:def 1
,WAYBEL_0:18;
  y = y "/\" y by YELLOW_0:25
    .= (y"/\").y by WAYBEL_1:def 18;
  hence thesis by A4,A5,FUNCT_1:def 6;
end;
