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;

theorem Th53:
  L is B_Lattice iff L.: is B_Lattice
proof
A1: L is distributive iff L.: is distributive by LATTICE2:50;
  L is C_Lattice iff L.: is C_Lattice by Th52;
  hence thesis by A1;
end;
