
theorem Th9:
  for X be set for Y be non empty Subset of InclPoset Filt
  BoolePoset X holds meet Y is Filter of BoolePoset X
proof
  let X be set;
  set L = BoolePoset X;
  let Y be non empty Subset of InclPoset Filt L;
A1: now
    let Z be set;
    assume Z in Y;
    then Z in the carrier of InclPoset Filt L;
    then Z in the carrier of RelStr(#Filt L, RelIncl (Filt L)#) by
YELLOW_1:def 1;
    then Z in the set of all  V where V is Filter of L  by
WAYBEL_0:def 24;
    then ex Z1 be Filter of L st Z1 = Z;
    hence Top L in Z by WAYBEL_4:22;
  end;
  Y c= the carrier of InclPoset Filt L;
  then
A2: Y c= the carrier of RelStr(#Filt L, RelIncl (Filt L)#) by YELLOW_1:def 1;
A3: for Z be Subset of L st Z in Y holds Z is upper
  proof
    let Z be Subset of L;
    assume Z in Y;
    then Z in Filt L by A2;
    then Z in the set of all  V where V is Filter of L  by
WAYBEL_0:def 24;
    then ex Z1 be Filter of L st Z1 = Z;
    hence thesis;
  end;
A4: now
    let V be Subset of L;
    assume V in Y;
    then V in Filt L by A2;
    then V in the set of all  Z where Z is Filter of L  by
WAYBEL_0:def 24;
    then
A5: ex V1 be Filter of L st V1 = V;
    hence V is upper;
    thus V is filtered by A5;
  end;
  Y c= bool the carrier of L
  proof
    let v be object;
    assume v in Y;
    then v in Filt L by A2;
    then v in the set of all  V where V is Filter of L  by
WAYBEL_0:def 24;
    then ex v1 be Filter of L st v1 = v;
    hence thesis;
  end;
  hence thesis by A1,A3,A4,SETFAM_1:def 1,YELLOW_2:37,39;
end;
