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 latt (L,I) is lower-bounded
proof
  set b9 = the Element of latt (L,I);
  reconsider b = b9 as Element of L by Th68;
  given c being Element of L such that
A1: for a being Element of L holds c"/\"a = c & a"/\"c = c;
A2: carr(latt (L,I)) = I by Th72;
  c"/\"b = c by A1;
  then reconsider c9 = c as Element of latt (L,I) by A2,Th22;
  take c9;
  let a9 be Element of latt (L,I);
  reconsider a = a9 as Element of L by Th68;
  thus c9"/\"a9 = c"/\"a by Th73
    .= c9 by A1;
  hence a9"/\"c9 = c9;
end;
