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 Th44:
  (X "\/" Y) \ (X "/\" Y) = (X \ Y) "\/" (Y \ X)
proof
  (X "\/" Y) \ (X "/\" Y) = (X "\/" Y) "/\" (X` "\/" Y`) by LATTICES:23
    .= ((X "\/" Y) "/\" X`) "\/" ((X "\/" Y) "/\" Y`) by LATTICES:def 11
    .= ((X "/\" X`) "\/" (Y "/\" X`)) "\/" ((X "\/" Y) "/\" Y`) by
LATTICES:def 11
    .= ((X "/\" X`) "\/" (Y "/\" X`)) "\/" ((X "/\" Y`) "\/" (Y "/\" Y`)) by
LATTICES:def 11
    .= (Bottom L "\/" (Y "/\" X`)) "\/" ((X "/\" Y`) "\/" (Y "/\" Y`)) by
LATTICES:20
    .= (Y "/\" X`) "\/" ((X "/\" Y`) "\/" Bottom L) by LATTICES:20
    .= (X \ Y) "\/" (Y \ X);
  hence thesis;
end;
