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' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' 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' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' 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' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))
let PA be a_partition of Y; :: thesis: Ex (a,PA,G) '<' 'not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (Ex (a,PA,G)) . z = TRUE or ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = TRUE )
A1: ('not' (Ex ((a '&' ('not' b)),PA,G))) . z = 'not' ((Ex ((a '&' ('not' b)),PA,G)) . z) by MARGREL1:def 19;
A2: ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = 'not' ((('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G)))) . z) by MARGREL1:def 19
.= 'not' ((('not' (Ex ((a '&' b),PA,G))) . z) '&' (('not' (Ex ((a '&' ('not' b)),PA,G))) . z)) by MARGREL1:def 20
.= 'not' (('not' ((Ex ((a '&' b),PA,G)) . z)) '&' ('not' ((Ex ((a '&' ('not' b)),PA,G)) . z))) by A1, MARGREL1:def 19 ;
assume A3: (Ex (a,PA,G)) . z = TRUE ; :: thesis: ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = TRUE

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;
hence contradiction by A3, BVFUNC_2:def 10; :: thesis: verum

end;

then consider x1 being Element of Y such that
A4: x1 in EqClass (z,(CompF (PA,G))) and
A5: a . x1 = TRUE ;
A6: (a '&' b) . x1 = TRUE '&' (b . x1) by A5, MARGREL1:def 20
.= b . x1 by MARGREL1:14 ;
A7: (a '&' ('not' b)) . x1 = TRUE '&' (('not' b) . x1) by A5, MARGREL1:def 20
.= ('not' b) . x1 by MARGREL1:14 ;

per cases ( b . x1 = TRUE or b . x1 = FALSE ) by XBOOLEAN:def 3;

suppose b . x1 = TRUE ; :: thesis: ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = TRUE

then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by A4, A6, BVFUNC_1:def 17;
hence ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = 'not' (('not' TRUE) '&' ('not' ((Ex ((a '&' ('not' b)),PA,G)) . z))) by A2, BVFUNC_2:def 10
.= 'not' (FALSE '&' ('not' ((Ex ((a '&' ('not' b)),PA,G)) . z))) by MARGREL1:11
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ;
:: thesis: verum

end;

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

then (a '&' ('not' b)) . x1 = 'not' FALSE by A7, MARGREL1:def 19;
then (a '&' ('not' b)) . x1 = TRUE by MARGREL1:11;
then (B_SUP ((a '&' ('not' b)),(CompF (PA,G)))) . z = TRUE by A4, BVFUNC_1:def 17;
hence ('not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))) . z = 'not' (('not' ((Ex ((a '&' b),PA,G)) . z)) '&' ('not' TRUE)) by A2, BVFUNC_2:def 10
.= 'not' (('not' ((Ex ((a '&' b),PA,G)) . z)) '&' FALSE) by MARGREL1:11
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ;
:: thesis: verum

end;

end;