let Y be non empty set ; :: thesis: for G being Subset of ()
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 (); :: 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 ;
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 ;
then (a '&' b) . x1 <> TRUE by ;
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 ;
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 = () 'or' FALSE by
.= FALSE 'or' FALSE by MARGREL1:11
.= FALSE ;
then (B_INF ((a 'imp' b),(CompF (PA,G)))) . z = FALSE by ;
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;