reserve L for Lattice;
reserve X,Y,Z,V for Element of L;
reserve L for D_Lattice;
reserve X,Y,Z for Element of L;
reserve L for 0_Lattice;
reserve X,Y,Z for Element of L;
reserve L for B_Lattice;
reserve X,Y,Z,V for Element of L;

theorem
  X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" (X "/\" Y "/\" Z)
proof
  X \ (Y \+\ Z) = X \ ((Y "\/" Z) \ (Y "/\" Z)) by Th64
    .= (X \ (Y "\/" Z)) "\/" (X "/\" (Y "/\" Z)) by Th37
    .= (X \ (Y "\/" Z)) "\/" (X "/\" Y "/\" Z) by LATTICES:def 7;
  hence thesis;
end;
