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) '&' (All b,PA,G) '<' Ex (a '&' 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) '&' (All b,PA,G) '<' Ex (a '&' 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) '&' (All b,PA,G) '<' Ex (a '&' b),PA,G
let PA be a_partition of Y; :: thesis: (Ex a,PA,G) '&' (All b,PA,G) '<' Ex (a '&' b),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((Ex a,PA,G) '&' (All b,PA,G)) . z = TRUE or (Ex (a '&' b),PA,G) . z = TRUE )
assume ((Ex a,PA,G) '&' (All b,PA,G)) . z = TRUE ; :: thesis: (Ex (a '&' b),PA,G) . z = TRUE
then A1: ((Ex a,PA,G) . z) '&' ((All b,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 . x = TRUE ) ; :: thesis: contradiction
then (B_INF b,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All b,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 . 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 A1, MARGREL1:45; :: thesis: verum

end;

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 = (a . x1) '&' (b . x1) by MARGREL1:def 21
.= TRUE '&' TRUE by A3, A4, A2
.= 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