let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for b, c, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G) '<' Ex (a '&' ('not' b)),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for b, c, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G) '<' Ex (a '&' ('not' b)),PA,G
let b, c, a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G) '<' Ex (a '&' ('not' b)),PA,G
let PA be a_partition of Y; :: thesis: (All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G) '<' Ex (a '&' ('not' b)),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G)) . z = TRUE or (Ex (a '&' ('not' b)),PA,G) . z = TRUE )
assume ((All (b 'imp' c),PA,G) '&' (Ex (a '&' ('not' c)),PA,G)) . z = TRUE ; :: thesis: (Ex (a '&' ('not' b)),PA,G) . z = TRUE
then A1: ((All (b 'imp' c),PA,G) . z) '&' ((Ex (a '&' ('not' c)),PA,G) . z) = TRUE by MARGREL1:def 21;

A2: now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not (b 'imp' c) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (b 'imp' c),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All (b 'imp' c),PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A1, MARGREL1:45; :: thesis: verum

end;

now

assume for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not (a '&' ('not' c)) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (a '&' ('not' c)),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
then (Ex (a '&' ('not' c)),PA,G) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A1, MARGREL1:45; :: thesis: verum

end;

then consider x1 being Element of Y such that
A3: ( x1 in EqClass z,(CompF PA,G) & (a '&' ('not' c)) . x1 = TRUE ) ;
(b 'imp' c) . x1 = TRUE by A2, A3;
then A4: ('not' (b . x1)) 'or' (c . x1) = TRUE by BVFUNC_1:def 11;
A5: ( 'not' (b . x1) = TRUE or 'not' (b . x1) = FALSE ) by XBOOLEAN:def 3;
A6: (a . x1) '&' (('not' c) . x1) = TRUE by A3, MARGREL1:def 21;

per cases ( ( a . x1 = TRUE & ('not' c) . x1 = TRUE & 'not' (b . x1) = TRUE ) or ( a . x1 = TRUE & ('not' c) . x1 = TRUE & c . x1 = TRUE ) ) by A4, A5, A6, BINARITH:7, MARGREL1:45;

suppose A7: ( a . x1 = TRUE & ('not' c) . x1 = TRUE & 'not' (b . x1) = TRUE ) ; :: thesis: (Ex (a '&' ('not' b)),PA,G) . z = TRUE

(a '&' ('not' b)) . x1 = (a . x1) '&' (('not' b) . x1) by MARGREL1:def 21
.= TRUE '&' TRUE by A7, MARGREL1:def 20
.= TRUE ;
then (B_SUP (a '&' ('not' b)),(CompF PA,G)) . z = TRUE by A3, BVFUNC_1:def 20;
hence (Ex (a '&' ('not' b)),PA,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum

end;

suppose A8: ( a . x1 = TRUE & ('not' c) . x1 = TRUE & c . x1 = TRUE ) ; :: thesis: (Ex (a '&' ('not' b)),PA,G) . z = TRUE

then 'not' (c . x1) = TRUE by MARGREL1:def 20;
hence (Ex (a '&' ('not' b)),PA,G) . z = TRUE by A8, MARGREL1:41; :: thesis: verum

end;

end;