reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem
  L is lower-bounded implies (.p.> = [#Bottom L,p#]
proof
  assume
A1: L is lower-bounded;
  let q;
A2: q in (.p.> iff q [= p by Th28;
  Bottom L [= p & Bottom L [= q by A1;
  hence thesis by A2,Th62;
end;
