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 (All a,PA,G) '&' (All b,PA,G) '<' a '&' b
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 (All a,PA,G) '&' (All b,PA,G) '<' a '&' b
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (All a,PA,G) '&' (All b,PA,G) '<' a '&' b
let PA be a_partition of Y; :: thesis: (All a,PA,G) '&' (All b,PA,G) '<' a '&' b
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All a,PA,G) '&' (All b,PA,G)) . z = TRUE or (a '&' b) . z = TRUE )
assume A1: ((All a,PA,G) '&' (All b,PA,G)) . z = TRUE ; :: thesis: (a '&' b) . z = TRUE
A2: ((All a,PA,G) '&' (All b,PA,G)) . z = ((All a,PA,G) . z) '&' ((All b,PA,G) . z) by MARGREL1:def 21;
A3: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

A4: now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) ; :: thesis: contradiction
then (B_INF a,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All a,PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A1, A2, MARGREL1:45; :: thesis: verum

end;

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, A2, MARGREL1:45; :: thesis: verum

end;

then A5: b . z = TRUE by A3;
thus (a '&' b) . z = (a . z) '&' (b . z) by MARGREL1:def 21
.= TRUE '&' TRUE by A3, A4, A5
.= TRUE ; :: thesis: verum