reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;
reserve f for UnOp of the carrier of 0L;
reserve 1L for upper-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 1L,
  b for Element of 1L;
reserve f,g for UnOp of the carrier of 1L;

theorem Th24:
  for F being ClosedSubset of 1L st Top 1L in F for B holds B c= F
  implies FinMeet B in F
proof
  let F be ClosedSubset of 1L;
  defpred X[Element of Fin the carrier of 1L] means $1 c= F implies FinMeet
  $1 in F;
A1: for B1 being Element of Fin the carrier of 1L for p being Element of
  1L st X[B1] holds X[B1 \/ {.p.}]
  proof
    let B1 be Element of Fin the carrier of 1L;
    let p be Element of 1L;
    assume
A2: B1 c=F implies FinMeet B1 in F;
    assume
A3: B1 \/ {p} c=F;
    then {p} c= F by XBOOLE_1:11;
    then p in F by ZFMISC_1:31;
    then FinMeet B1 "/\" p in F by A2,A3,LATTICES:def 24,XBOOLE_1:11;
    hence thesis by Th21;
  end;
  assume Top 1L in F;
  then
A4: X[{}.the carrier of 1L] by Lm2;
  thus for B being Element of Fin the carrier of 1L holds X[B] from SETWISEO:
  sch 4(A4,A1);
end;
