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 Th70:
  latt (L,P) = (latt (L.:,P.:)).:
proof
  (ex o1, o2 st o1 = join(L)||P & o2 = met(L)||P & latt (L,P) = LattStr
  (#P, o1, o2#) )& ex o3, o4 being BinOp of P.: st o3 = join(L.:)||(P.:) & o4 =
  met (L.:)||(P.:) & latt (L.:,P.:) = LattStr (#P.:, o3, o4#) by Def14;
  hence thesis;
end;
