let L be B_Lattice; :: thesis: for X, Y, Z being Element of L holds X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" ((X "/\" Y) "/\" Z)
let X, Y, Z be Element of L; :: thesis: X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" ((X "/\" Y) "/\" Z)
X \ (Y \+\ Z) = X \ ((Y "\/" Z) \ (Y "/\" Z)) by Th64
.= (X \ (Y "\/" Z)) "\/" (X "/\" (Y "/\" Z)) by Th37
.= (X \ (Y "\/" Z)) "\/" ((X "/\" Y) "/\" Z) by LATTICES:def 7 ;
hence X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" ((X "/\" Y) "/\" Z) ; :: thesis: verum