
theorem Th3:
  for R being antisymmetric with_infima transitive non empty RelStr
, x, y being Element of R holds downarrow (x"/\"y) = (downarrow x) /\ downarrow
  y
proof
  let R be antisymmetric with_infima transitive non empty RelStr, x,y be
  Element of R;
  now
    let z be object;
    hereby
      assume
A1:   z in downarrow (x"/\"y);
      then reconsider z9 = z as Element of R;
A2:   z9 <= (x"/\"y) by A1,WAYBEL_0:17;
      (x"/\"y) <= y by YELLOW_0:23;
      then z9 <= y by A2,YELLOW_0:def 2;
      then
A3:   z9 in downarrow y by WAYBEL_0:17;
      (x"/\"y) <= x by YELLOW_0:23;
      then z9 <= x by A2,YELLOW_0:def 2;
      then z9 in downarrow x by WAYBEL_0:17;
      hence z in (downarrow x) /\ downarrow y by A3,XBOOLE_0:def 4;
    end;
    assume
A4: z in (downarrow x) /\ downarrow y;
    then reconsider z9 = z as Element of R;
    z in downarrow y by A4,XBOOLE_0:def 4;
    then
A5: z9 <= y by WAYBEL_0:17;
    z in (downarrow x) by A4,XBOOLE_0:def 4;
    then z9 <= x by WAYBEL_0:17;
    then x"/\"y >= z9 by A5,YELLOW_0:23;
    hence z in downarrow (x"/\"y) by WAYBEL_0:17;
  end;
  hence thesis by TARSKI:2;
end;
