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)` = (X "/\" Y) "\/" (X` "/\" Y`)
proof
  thus (X \+\ Y)` = (X \ Y)` "/\" (Y \ X)` by LATTICES:24
    .=(X` "\/" Y``) "/\" (Y "/\" X`)` by LATTICES:23
    .=(X` "\/" Y``) "/\" (Y` "\/" X``) by LATTICES:23
    .=(X` "\/" Y) "/\" (Y` "\/" X``)
    .=(X` "\/" Y) "/\" (Y` "\/" X)
    .=(X` "/\" (Y` "\/" X)) "\/" (Y "/\" (Y` "\/" X)) by LATTICES:def 11
    .=((X` "/\" Y`) "\/" (X` "/\" X)) "\/" (Y "/\" (Y` "\/" X)) by
LATTICES:def 11
    .=((X` "/\" Y`) "\/" (X` "/\" X)) "\/" ((Y "/\" Y`) "\/" (Y "/\" X)) by
LATTICES:def 11
    .=((X` "/\" Y`) "\/" Bottom L) "\/" ((Y "/\" Y`) "\/" (Y "/\" X)) by
LATTICES:20
    .=(X` "/\" Y`) "\/" (Bottom L "\/" (Y "/\" X)) by LATTICES:20
    .=(X "/\" Y) "\/" (X` "/\" Y`);
end;
