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) "/\" (X \ Z)
proof
  thus X \ (Y "\/" Z) = X "/\" (Y` "/\" Z`) by LATTICES:24
    .= (X "/\" X "/\" Y`) "/\" Z` by LATTICES:def 7
    .= (X "/\" (X "/\" Y`)) "/\" Z` by LATTICES:def 7
    .= (X \ Y) "/\" (X \ Z) by LATTICES:def 7;
end;
