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)
proof
  X "\/" Y = (X "\/" Y) "/\" Top L
    .= (X "\/" Y) "/\" (X "\/" X`) by LATTICES:21
    .= X "\/" (Y "/\" X`) by LATTICES:11
    .= ((X "/\" Y`) "\/" (X "/\" X)) "\/" (Y "/\" X`) by LATTICES:def 8
    .= ((X "/\" Y`) "\/" (X "/\" X``)) "\/" (Y "/\" (X` "/\" X`))
    .= (X "/\" (Y` "\/" X``)) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:def 11
    .= (X "/\" (Y "/\" X`)`) "\/" (Y "/\" (X` "/\" X`)) by LATTICES:23
    .= X \+\ (Y \ X) by LATTICES:def 7;
  hence thesis;
end;
