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
  latt (L,(.L.>) = the LattStr of L & latt (L,<.L.)) = the LattStr of L
proof
A1: dom met(L) = [:carr(L), carr(L):] & join(L)|(dom join(L) qua set) = join
  (L) by FUNCT_2:def 1,RELAT_1:68;
  (ex o1,o2 being BinOp of (.L.> st o1 = (the L_join of L)|| (.L.> & o2 =
  (the L_meet of L)||(.L.> & latt (L,(.L.>) = LattStr (#(.L.>, o1, o2#) )& dom
  join( L) = [:carr(L), carr(L):] by Def14,FUNCT_2:def 1;
  hence thesis by A1,RELAT_1:68;
end;
