let Y be non empty set ; :: thesis: 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); :: thesis: 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; :: thesis: 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; :: thesis: (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( 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 ; :: thesis: ('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;

now :: thesis: ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE )

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (a,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;
then (Ex (a,PA,G)) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A1, MARGREL1:12; :: thesis: verum

end;

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;

per cases ( a . x1 = FALSE or b . x1 = FALSE ) by A4, MARGREL1:12;

suppose a . x1 = FALSE ; :: thesis: ('not' (All ((a 'imp' b),PA,G))) . z = TRUE

hence ('not' (All ((a 'imp' b),PA,G))) . z = TRUE by A3; :: thesis: verum

end;

suppose b . x1 = FALSE ; :: thesis: ('not' (All ((a 'imp' b),PA,G))) . z = TRUE

then (a 'imp' b) . x1 = ('not' TRUE) 'or' FALSE by A3, BVFUNC_1:def 8
.= FALSE 'or' FALSE by MARGREL1:11
.= FALSE ;
then (B_INF ((a 'imp' b),(CompF (PA,G)))) . z = FALSE by A2, BVFUNC_1:def 16;
then (All ((a 'imp' b),PA,G)) . z = FALSE by BVFUNC_2:def 9;
hence ('not' (All ((a 'imp' b),PA,G))) . z = 'not' FALSE by MARGREL1:def 19
.= TRUE by MARGREL1:11 ;
:: thesis: verum

end;

end;