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;

theorem Th17:
  for A being ClosedSubset of 0L st Bottom 0L in A for B holds
  B c= A implies FinJoin B in A
proof
  let A be ClosedSubset of 0L;
  defpred X[Element of Fin the carrier of 0L] means $1 c= A implies FinJoin
  $1 in A;
A1: for B1 being Element of Fin the carrier of 0L for p being Element of
  0L st X[B1] holds X[B1 \/ {.p.}]
  proof
    let B1 be Element of Fin the carrier of 0L;
    let p be Element of 0L;
    assume
A2: B1 c=A implies FinJoin B1 in A;
    assume
A3: B1 \/ {p} c=A;
    then {p} c= A by XBOOLE_1:11;
    then p in A by ZFMISC_1:31;
    then FinJoin B1 "\/" p in A by A2,A3,LATTICES:def 25,XBOOLE_1:11;
    hence thesis by Th14;
  end;
  assume Bottom 0L in A;
  then
A4: X[{}.the carrier of 0L] by Lm1;
  thus for B being Element of Fin the carrier of 0L holds X[B] from SETWISEO:
  sch 4(A4,A1);
end;
