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
  (X \+\ Y) \+\ (X "\/" Y) = ((X \+\ Y) "/\" (X` "/\" Y`)) "\/" ((X "\/" Y
  ) \ (X \+\ Y)) by LATTICES:24
    .= ( ((X "/\" Y`) "/\" (X` "/\" Y`)) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)
  ) ) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 11
    .= ( (X "/\" (Y` "/\" (Y` "/\" X`))) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)
  ) ) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
    .= ( (X "/\" ((Y` "/\" Y`) "/\" X`)) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)
  ) ) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
    .= ( ((X "/\" X`) "/\" Y`) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) ) "\/" (
  (X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
    .= ( (Bottom L "/\" Y`) "\/" ((Y "/\" X`) "/\" (X` "/\" Y`)) ) "\/" ((X
  "\/" Y) \ (X \+\ Y)) by LATTICES:20
    .= (Y "/\" (X` "/\" (X` "/\" Y`))) "\/" ((X "\/" Y) \ (X \+\ Y)) by
LATTICES:def 7
    .= (Y "/\" ((X` "/\" X`) "/\" Y`)) "\/" ((X "\/" Y) \ (X \+\ Y)) by
LATTICES:def 7
    .= ((Y "/\" Y`) "/\" X`) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:def 7
    .= (Bottom L "/\" X`) "\/" ((X "\/" Y) \ (X \+\ Y)) by LATTICES:20
    .= Y "/\" X by Lm1;
  hence thesis;
end;
