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;
