let L be B_Lattice; :: thesis: for X, Y being Element of L st X misses Y holds
(X "\/" Y) \ Y = X
let X, Y be Element of L; :: thesis: ( X misses Y implies (X "\/" Y) \ Y = X )
assume X "/\" Y = Bottom L ; :: according to BOOLEALG:def 4 :: thesis: (X "\/" Y) \ Y = X
then (X `) "\/" (X "/\" Y) = X ` ;
then ((X `) "\/" X) "/\" ((X `) "\/" Y) = X ` by LATTICES:11;
then (Top L) "/\" ((X `) "\/" Y) = X ` by LATTICES:21;
then ((X `) "\/" Y) ` = X ;
then A1: ((X `) `) "/\" (Y `) = X by LATTICES:24;
(X "\/" Y) \ Y = (X "/\" (Y `)) "\/" (Y "/\" (Y `)) by LATTICES:def 11
.= (X "/\" (Y `)) "\/" (Bottom L) by LATTICES:20
.= X "/\" (Y `) ;
hence (X "\/" Y) \ Y = X by A1; :: thesis: verum