let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a '&' b),PA,G '<' a '&' (All b,PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a '&' b),PA,G '<' a '&' (All b,PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y holds All (a '&' b),PA,G '<' a '&' (All b,PA,G)
let PA be a_partition of Y; :: thesis: All (a '&' b),PA,G '<' a '&' (All b,PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (a '&' b),PA,G) . z = TRUE or (a '&' (All b,PA,G)) . z = TRUE )
assume A1: (All (a '&' b),PA,G) . z = TRUE ; :: thesis: (a '&' (All b,PA,G)) . z = TRUE
A2: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ; :: thesis: contradiction
then consider x1 being Element of Y such that
A3: ( x1 in EqClass z,(CompF PA,G) & b . x1 <> TRUE ) ;
A4: b . x1 = FALSE by A3, XBOOLEAN:def 3;
(a '&' b) . x1 = TRUE by A1, A3, BVFUNC_1:def 19;
then (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 21;
hence contradiction by A4; :: thesis: verum

end;

then A5: (B_INF b,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 19;

now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) ; :: thesis: contradiction
then consider x1 being Element of Y such that
A6: ( x1 in EqClass z,(CompF PA,G) & a . x1 <> TRUE ) ;
A7: a . x1 = FALSE by A6, XBOOLEAN:def 3;
(a '&' b) . x1 = TRUE by A1, A6, BVFUNC_1:def 19;
then (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 21;
hence contradiction by A7; :: thesis: verum

end;

then a . z = TRUE by A2;
then (a '&' (All b,PA,G)) . z = TRUE '&' TRUE by A5, MARGREL1:def 21
.= TRUE ;
hence (a '&' (All b,PA,G)) . z = TRUE ; :: thesis: verum