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 Th29:
  X \ Y = Bottom L iff X [= Y
proof
  thus X \ Y = Bottom L implies X [= Y
  proof
A1: X "/\" Y` = Bottom L implies X [= Y`` by LATTICES:25;
    assume X \ Y = Bottom L;
    hence thesis by A1;
  end;
  assume X [= Y;
  then X "/\" Y` [= Y` "/\" Y by LATTICES:9;
  then
A2: X "/\" Y` [= Bottom L by LATTICES:20;
  Bottom L [= X \ Y by LATTICES:16;
  hence thesis by A2;
end;
