let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
let G be Subset of (PARTITIONS Y); for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
let a, b be Function of Y,BOOLEAN; for PA being a_partition of Y holds (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
let PA be a_partition of Y; (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
let z be Element of Y; BVFUNC_1:def 12 ( not ((Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G)))) . z = TRUE or ('not' (All ((a 'imp' b),PA,G))) . z = TRUE )
assume
((Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G)))) . z = TRUE
; ('not' (All ((a 'imp' b),PA,G))) . z = TRUE
then A1:
((Ex (a,PA,G)) . z) '&' (('not' (Ex ((a '&' b),PA,G))) . z) = TRUE
by MARGREL1:def 20;
then consider x1 being Element of Y such that
A2:
x1 in EqClass (z,(CompF (PA,G)))
and
A3:
a . x1 = TRUE
;
('not' (Ex ((a '&' b),PA,G))) . z = TRUE
by A1, MARGREL1:12;
then
'not' ((Ex ((a '&' b),PA,G)) . z) = TRUE
by MARGREL1:def 19;
then
( Ex ((a '&' b),PA,G) = B_SUP ((a '&' b),(CompF (PA,G))) & (Ex ((a '&' b),PA,G)) . z = FALSE )
by BVFUNC_2:def 10, MARGREL1:11;
then
(a '&' b) . x1 <> TRUE
by A2, BVFUNC_1:def 17;
then
(a '&' b) . x1 = FALSE
by XBOOLEAN:def 3;
then A4:
(a . x1) '&' (b . x1) = FALSE
by MARGREL1:def 20;