reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;

theorem Th27:
  L is B_Lattice implies for p,q holds p "/\" (p` "\/" q) [= q &
  for r st p "/\" r [= q holds r [= p` "\/" q
proof
  assume L is B_Lattice;
  then reconsider S = L as B_Lattice;
  reconsider J = S as 1_Lattice;
  reconsider K = S as 0_Lattice;
  let p,q;
  set r = p` "\/" q;
  reconsider p9 = p, q9 = q as Element of K;
  reconsider p99 = p as Element of S;
A1: p99 "/\" p99` = Bottom L & Bottom K "\/" (p9 "/\" q9) = p9 "/\" q9 by
LATTICES:20;
  reconsider K = S as D_Lattice;
  reconsider p9 = p, q9 = q, r9 = r as Element of K;
  p9 "/\" r9 = (p9 "/\" p9`) "\/" (p9 "/\" q9) by LATTICES:def 11;
  hence p "/\" r [= q by A1,LATTICES:6;
  let r1;
  reconsider r19 = r1 as Element of K;
  reconsider pp = p, r99 = r1 as Element of J;
A2: p99` "\/" p99 = Top L & Top J "/\" (pp` "\/" r99) = pp` "\/" r99 by
LATTICES:21;
  assume p "/\" r1 [= q;
  then
A3: p` "\/" (p "/\" r1) [= r by Th1;
  p9` "\/" (p9 "/\" r19) = (p9` "\/" p9) "/\" (p9` "\/" r19) & r1 [= r1
  "\/" p ` by LATTICES:5,11;
  hence thesis by A3,A2,LATTICES:7;
end;
