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 Th79:
  Top latt (L,(.p.>) = p
proof
  latt (L,(.p.>) = (latt (L.:,(.p.>.:)).: by Th70
    .= (latt (L.:,<.p.:.))).: by Th29
    .= (latt <.p.:.)).: by Th69;
  hence Top latt (L,(.p.>) = Bottom latt <.p.:.) by LATTICE2:61
    .= p by FILTER_0:56;
end;
