let Y be non empty set ; :: thesis: for a, b being Function of 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 Function of 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 12 :: 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: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE

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
A3: x1 in EqClass (z,(CompF (PA,G))) and
A4: a . x1 <> TRUE ;
(a '&' b) . x1 = TRUE by A1, A3, BVFUNC_1:def 16;
then A5: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;
a . x1 = FALSE by A4, XBOOLEAN:def 3;
hence contradiction by A5; :: thesis: verum

end;

now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE

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
A6: x1 in EqClass (z,(CompF (PA,G))) and
A7: b . x1 <> TRUE ;
(a '&' b) . x1 = TRUE by A1, A6, BVFUNC_1:def 16;
then A8: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;
b . x1 = FALSE by A7, XBOOLEAN:def 3;
hence contradiction by A8; :: thesis: verum

end;

then A9: (B_INF (b,(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
then a . z = TRUE by A2;
then (a '&' (All (b,PA,G))) . z = TRUE '&' TRUE by A9, MARGREL1:def 20
.= TRUE ;
hence (a '&' (All (b,PA,G))) . z = TRUE ; :: thesis: verum