let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for c, b, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds ((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for c, b, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds ((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),PA,G
let c, b, a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds ((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),PA,G
let PA be a_partition of Y; :: thesis: ((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G)) . z = TRUE or (Ex (a '&' b),PA,G) . z = TRUE )
assume (((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G)) . z = TRUE ; :: thesis: (Ex (a '&' b),PA,G) . z = TRUE
then A1: (((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) . z) '&' ((All (c 'imp' a),PA,G) . z) = TRUE by MARGREL1:def 21;
then (((Ex c,PA,G) . z) '&' ((All (c 'imp' b),PA,G) . z)) '&' ((All (c 'imp' a),PA,G) . z) = TRUE by MARGREL1:def 21;
then A2: ((Ex c,PA,G) . z) '&' ((All (c 'imp' b),PA,G) . z) = TRUE by MARGREL1:45;

now

assume for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not c . x = TRUE ) ; :: thesis: contradiction
then (B_SUP c,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
then (Ex c,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
A3: x1 in EqClass z,(CompF PA,G) and
A4: c . x1 = TRUE ;
A5: ( 'not' (c . x1) = TRUE or 'not' (c . x1) = FALSE ) by XBOOLEAN:def 3;

now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not (c 'imp' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (c 'imp' b),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All (c 'imp' b),PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A2, MARGREL1:45; :: thesis: verum

end;

then (c 'imp' b) . x1 = TRUE by A3;
then A6: ('not' (c . x1)) 'or' (b . x1) = TRUE by BVFUNC_1:def 11;

now

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

end;

then (c 'imp' a) . x1 = TRUE by A3;
then A7: ('not' (c . x1)) 'or' (a . x1) = TRUE by BVFUNC_1:def 11;

per cases ( 'not' (c . x1) = TRUE or ( 'not' (c . x1) = TRUE & b . x1 = TRUE ) or ( a . x1 = TRUE & 'not' (c . x1) = TRUE ) or ( a . x1 = TRUE & b . x1 = TRUE ) ) by A7, A5, A6, BINARITH:7;

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

hence (Ex (a '&' b),PA,G) . z = TRUE by A4, MARGREL1:41; :: thesis: verum

end;

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

hence (Ex (a '&' b),PA,G) . z = TRUE by A4, MARGREL1:41; :: thesis: verum

end;

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

hence (Ex (a '&' b),PA,G) . z = TRUE by A4, MARGREL1:41; :: thesis: verum

end;

suppose ( a . x1 = TRUE & b . x1 = TRUE ) ; :: thesis: (Ex (a '&' b),PA,G) . z = TRUE

then (a '&' b) . x1 = TRUE '&' TRUE by MARGREL1:def 21
.= TRUE ;
then (B_SUP (a '&' b),(CompF PA,G)) . z = TRUE by A3, BVFUNC_1:def 20;
hence (Ex (a '&' b),PA,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum

end;

end;