
theorem Th7:
  for X be set holds BoolePoset X is arithmetic
proof
  let X be set;
  now
    let x,y be Element of CompactSublatt BoolePoset X;
    reconsider x9 = x, y9 = y as Element of BoolePoset X by YELLOW_0:58;
    x9 is compact by WAYBEL_8:def 1;
    then x9 is finite by WAYBEL_8:28;
    then x9 /\ y9 is finite;
    then x9 "/\" y9 is finite by YELLOW_1:17;
    then x9 "/\" y9 is compact by WAYBEL_8:28;
    then reconsider
xy = x9 "/\" y9 as Element of CompactSublatt BoolePoset X by WAYBEL_8:def 1;
A1: now
      let z be Element of CompactSublatt BoolePoset X;
      reconsider z9 = z as Element of BoolePoset X by YELLOW_0:58;
      assume that
A2:   z <= x and
A3:   z <= y;
      z9 <= y9 by A3,YELLOW_0:59;
      then
A4:   z9 c= y9 by YELLOW_1:2;
      z9 <= x9 by A2,YELLOW_0:59;
      then z9 c= x9 by YELLOW_1:2;
      then z9 c= x9 /\ y9 by A4,XBOOLE_1:19;
      then z9 c= x9 "/\" y9 by YELLOW_1:17;
      then z9 <= x9 "/\" y9 by YELLOW_1:2;
      hence xy >= z by YELLOW_0:60;
    end;
    x9 /\ y9 c= y9 by XBOOLE_1:17;
    then x9 "/\" y9 c= y9 by YELLOW_1:17;
    then x9 "/\" y9 <= y9 by YELLOW_1:2;
    then
A5: xy <= y by YELLOW_0:60;
    x9 /\ y9 c= x9 by XBOOLE_1:17;
    then x9 "/\" y9 c= x9 by YELLOW_1:17;
    then x9 "/\" y9 <= x9 by YELLOW_1:2;
    then xy <= x by YELLOW_0:60;
    hence ex_inf_of {x,y},CompactSublatt BoolePoset X by A5,A1,YELLOW_0:19;
  end;
  then CompactSublatt BoolePoset X is with_infima by YELLOW_0:21;
  hence thesis by WAYBEL_8:19;
end;
