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' (('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 Element of Funcs 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 Element of Funcs 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 15 :: 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 20;
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 20
.= 'not' ((('not' (Ex (a '&' b),PA,G)) . z) '&' (('not' (Ex (a '&' ('not' b)),PA,G)) . z)) by MARGREL1:def 21
.= 'not' (('not' ((Ex (a '&' b),PA,G) . z)) '&' ('not' ((Ex (a '&' ('not' b)),PA,G) . z))) by A1, MARGREL1:def 20 ;
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

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;
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 21
.= b . x1 by MARGREL1:50 ;
A7: (a '&' ('not' b)) . x1 = TRUE '&' (('not' b) . x1) by A5, MARGREL1:def 21
.= ('not' b) . x1 by MARGREL1:50 ;

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 20;
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 MARGREL1:41
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ;
:: 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 20;
then (a '&' ('not' b)) . x1 = TRUE by MARGREL1:41;
then (B_SUP (a '&' ('not' b)),(CompF PA,G)) . z = TRUE by A4, BVFUNC_1:def 20;
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 MARGREL1:41
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ;
:: thesis: verum

end;

end;