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
  set XY = X \+\ Y, A = Y "/\" X`;
  XY \+\ (X "/\" Y) = (XY "/\" (X` "\/" Y`)) "\/" ((X "/\" Y) "/\" XY`) by
LATTICES:23
    .= ( (XY "/\" X`) "\/" (XY "/\" Y`) ) "\/" ((X "/\" Y) "/\" XY`) by
LATTICES:def 11
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" ((X "/\" Y`)` "/\" A
  `) ) by LATTICES:24
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" ((X` "\/" Y``) "/\"
  A`) ) by LATTICES:23
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" ((X` "\/" Y``) "/\"
  (Y` "\/" X``)) ) by LATTICES:23
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" ((X` "\/" Y) "/\" (Y
  ` "\/" X``)) )
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Y) "/\" ((X` "\/" Y) "/\" (Y
  ` "\/" X)) )
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" (((X "/\" Y) "/\" (X` "\/" Y)) "/\" (
  Y` "\/" X)) by LATTICES:def 7
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ( (((X "/\" Y) "/\" X`) "\/" ((X "/\"
  Y) "/\" Y)) "/\" (Y` "\/" X)) by LATTICES:def 11
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ( ((Y "/\" (X "/\" X`)) "\/" ((X "/\"
  Y) "/\" Y)) "/\" (Y` "\/" X)) by LATTICES:def 7
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ( ((Y "/\" Bottom L) "\/" ((X "/\" Y)
  "/\" Y)) "/\" (Y` "\/" X)) by LATTICES:20
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" (Y "/\" Y)) "/\" (Y` "\/" X)
  ) by LATTICES:def 7
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" (((X "/\" Y) "/\" Y`) "\/" ((X "/\" Y
  ) "/\" X)) by LATTICES:def 11
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" (Y "/\" Y`)) "\/" ((X "/\" Y
  ) "/\" X)) by LATTICES:def 7
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" ((X "/\" Bottom L) "\/" ((X "/\" Y)
  "/\" X)) by LATTICES:20
    .= ( (XY \ X) "\/" (XY \ Y) ) "\/" (Y "/\" (X "/\" X)) by LATTICES:def 7
    .= ( (((X "/\" Y`) "/\" X`) "\/" (A "/\" X`)) "\/" (((X \ Y) "\/" (Y \ X
  )) "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:def 11
    .= ( ((Y` "/\" (X "/\" X`)) "\/" (A "/\" X`)) "\/" (((X \ Y) "\/" (Y \ X
  )) "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:def 7
    .= ( ((Y` "/\" (X "/\" X`)) "\/" (Y "/\" (X` "/\" X`))) "\/" (((X \ Y)
  "\/" (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:def 7
    .= ( ((Y` "/\" Bottom L) "\/" (Y "/\" (X` "/\" X`))) "\/" (((X \ Y) "\/"
  (Y \ X)) "/\" Y`) ) "\/" (Y "/\" X) by LATTICES:20
    .= ( A "\/" (((X "/\" Y`) "/\" Y`) "\/" (A "/\" Y`)) ) "\/" (Y "/\" X)
  by LATTICES:def 11
    .= ( A "\/" ( (X "/\" (Y` "/\" Y`)) "\/" (A "/\" Y`)) ) "\/" (Y "/\" X)
  by LATTICES:def 7
    .= ( A "\/" ( (X "/\" Y`) "\/" (X` "/\" (Y "/\" Y`))) ) "\/" (Y "/\" X)
  by LATTICES:def 7
    .= ( A "\/" ((X "/\" Y`) "\/" (X` "/\" Bottom L) )) "\/" (Y "/\" X) by
LATTICES:20
    .= A "\/" ( (X "/\" Y`) "\/" (Y "/\" X) ) by LATTICES:def 5
    .= A "\/" (X "/\" (Y` "\/" Y) ) by LATTICES:def 11
    .= A "\/" (X "/\" Top L) by LATTICES:21
    .= (Y "\/" X) "/\" (X` "\/" X ) by LATTICES:11
    .= (Y "\/" X) "/\" Top L by LATTICES:21
    .= Y "\/" X;
  hence thesis;
end;
