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 = Bottom L iff X \ Y = X
proof
  thus X "/\" Y = Bottom L implies X \ Y = X
  proof
    assume X "/\" Y = Bottom L;
    then
A1:  X "/\" X [= X "/\" Y` by LATTICES:9,25;
    X \ Y [= X by LATTICES:6;
    hence thesis by A1;
  end;
  assume X \ Y = X;
  then X` "\/" Y`` = X` by LATTICES:23;
  then X "/\" (X` "\/" Y) [= X "/\" X`;
  then (X "/\" X`) "\/" (X "/\" Y) [= X "/\" X` by LATTICES:def 11;
  then Bottom L "\/" (X "/\" Y) [= X "/\" X` by LATTICES:20;
  then
A2: X "/\" Y [= Bottom L by LATTICES:20;
  Bottom L [= X "/\" Y by LATTICES:16;
  hence thesis by A2;
end;
