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 Th80:
  L is lower-bounded implies latt (L,(.p.>) is lower-bounded &
  Bottom latt (L,(.p.>) = Bottom L
proof
  assume
A1: L is lower-bounded;
  then
A2: L.: is upper-bounded by LATTICE2:48;
  then
A3: latt <.p.:.) is upper-bounded by FILTER_0:52;
A4: latt (L,(.p.>) = (latt (L.:,(.p.>.:)).: by Th70
    .= (latt (L.:,<.p.:.))).: by Th29
    .= (latt <.p.:.)).: by Th69;
  hence latt (L,(.p.>) is lower-bounded by A3,LATTICE2:49;
  Top latt <.p.:.) = Top (L.:) by A2,FILTER_0:57;
  hence Bottom latt (L,(.p.>) = Top (L.:) by A4,A3,LATTICE2:62
    .= Bottom L by A1,LATTICE2:61;
end;
