let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds Ex a,PA,G '<' 'not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds Ex a,PA,G '<' 'not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds Ex a,PA,G '<' 'not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))
let PA be a_partition of Y; :: thesis: Ex a,PA,G '<' 'not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (Ex a,PA,G) . z = TRUE or ('not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))) . z = TRUE )
assume A1: (Ex a,PA,G) . z = TRUE ; :: thesis: ('not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))) . z = TRUE

now

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 20;
then (Ex a,PA,G) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A1; :: thesis: verum

end;

then consider x1 being Element of Y such that
A2: ( x1 in EqClass z,(CompF PA,G) & a . x1 = TRUE ) ;
A3: ('not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))) . z = 'not' (((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G)) . z) by MARGREL1:def 20
.= 'not' (((All (a 'imp' b),PA,G) . z) '&' ((All (a 'imp' ('not' b)),PA,G) . z)) by MARGREL1:def 21 ;
A4: (a 'imp' b) . x1 = ('not' TRUE ) 'or' (b . x1) by A2, BVFUNC_1:def 11
.= FALSE 'or' (b . x1) by MARGREL1:41
.= b . x1 by BINARITH:7 ;
A5: (a 'imp' ('not' b)) . x1 = ('not' TRUE ) 'or' (('not' b) . x1) by A2, BVFUNC_1:def 11
.= FALSE 'or' (('not' b) . x1) by MARGREL1:41
.= ('not' b) . x1 by BINARITH:7 ;

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

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

then (a 'imp' ('not' b)) . x1 = 'not' TRUE by A5, MARGREL1:def 20
.= FALSE by MARGREL1:41 ;
then (B_INF (a 'imp' ('not' b)),(CompF PA,G)) . z = FALSE by A2, BVFUNC_1:def 19;
hence ('not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))) . z = 'not' (((All (a 'imp' b),PA,G) . z) '&' FALSE ) by A3, BVFUNC_2:def 9
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ;
:: thesis: verum

end;

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

then (B_INF (a 'imp' b),(CompF PA,G)) . z = FALSE by A2, A4, BVFUNC_1:def 19;
hence ('not' ((All (a 'imp' b),PA,G) '&' (All (a 'imp' ('not' b)),PA,G))) . z = 'not' (FALSE '&' ((All (a 'imp' ('not' b)),PA,G) . z)) by A3, BVFUNC_2:def 9
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ;
:: thesis: verum

end;

end;