
theorem
  for X be set for Y be non empty Subset of InclPoset Filt BoolePoset X
  holds ex_inf_of Y,InclPoset Filt BoolePoset X & "/\"(Y,InclPoset Filt
  BoolePoset X) = meet Y
proof
  let X be set;
  set L = InclPoset Filt BoolePoset X;
  let Y be non empty Subset of L;
  meet Y is Filter of BoolePoset X by Th9;
  then meet Y in the set of all  Z where Z is Filter of BoolePoset X;
  then meet Y in the carrier of RelStr(#Filt BoolePoset X, RelIncl (Filt
    BoolePoset X)#) by WAYBEL_0:def 24;
  then reconsider a = meet Y as Element of L by YELLOW_1:def 1;
A1: for b be Element of L st b is_<=_than Y holds b <= a
  proof
    let b be Element of L;
    assume
A2: b is_<=_than Y;
    for x being set st x in Y holds b c= x by YELLOW_1:3,A2,LATTICE3:def 8;
    then b c= meet Y by SETFAM_1:5;
    hence thesis by YELLOW_1:3;
  end;
  for b being Element of L st b in Y holds a <= b by YELLOW_1:3,SETFAM_1:3;
  then a is_<=_than Y by LATTICE3:def 8;
  hence thesis by A1,YELLOW_0:31;
end;
