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' (Ex (a '&' b),PA,G)) '<' 'not' (All (a 'imp' 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' (Ex (a '&' b),PA,G)) '<' 'not' (All (a 'imp' 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' (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)
A1: Ex (a '&' b),PA,G = B_SUP (a '&' b),(CompF PA,G) by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: 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 A2: ((Ex a,PA,G) . z) '&' (('not' (Ex (a '&' b),PA,G)) . z) = TRUE by MARGREL1:def 21;
then ( (Ex a,PA,G) . z = TRUE & ('not' (Ex (a '&' b),PA,G)) . z = TRUE ) by MARGREL1:45;
then 'not' ((Ex (a '&' b),PA,G) . z) = TRUE by MARGREL1:def 20;
then A3: (Ex (a '&' b),PA,G) . z = FALSE by MARGREL1:41;

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 A2, MARGREL1:45; :: thesis: verum

end;

then consider x1 being Element of Y such that
A4: ( x1 in EqClass z,(CompF PA,G) & a . x1 = TRUE ) ;
(a '&' b) . x1 <> TRUE by A1, A3, A4, BVFUNC_1:def 20;
then (a '&' b) . x1 = FALSE by XBOOLEAN:def 3;
then A5: (a . x1) '&' (b . x1) = FALSE by MARGREL1:def 21;

per cases ( a . x1 = FALSE or b . x1 = FALSE ) by A5, MARGREL1:45;

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 A4; :: 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 A4, BVFUNC_1:def 11
.= FALSE 'or' FALSE by MARGREL1:41
.= FALSE ;
then (B_INF (a 'imp' b),(CompF PA,G)) . z = FALSE by A4, BVFUNC_1:def 19;
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 20
.= TRUE by MARGREL1:41 ;
:: thesis: verum

end;

end;