let Y be non empty set ; :: thesis: for G being Subset of ()
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G)) '<' Ex ((a '&' b),PA,G)

let G be Subset of (); :: thesis: for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G)) '<' Ex ((a '&' b),PA,G)

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G)) '<' Ex ((a '&' b),PA,G)
let PA be a_partition of Y; :: thesis: (Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G)) '<' Ex ((a '&' b),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G))) . z = TRUE or (Ex ((a '&' b),PA,G)) . z = TRUE )
A1: 'not' FALSE = TRUE by MARGREL1:11;
assume ((Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G))) . z = TRUE ; :: thesis: (Ex ((a '&' b),PA,G)) . z = TRUE
then A2: ((Ex (a,PA,G)) . z) '&' ((All ((a 'imp' b),PA,G)) . z) = TRUE by MARGREL1:def 20;
now :: thesis: ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE )
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 17;
then (Ex (a,PA,G)) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A2, MARGREL1:12; :: 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 ;
now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' b) . x = TRUE
assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not (a 'imp' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF ((a 'imp' b),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
then (All ((a 'imp' b),PA,G)) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A2, MARGREL1:12; :: thesis: verum
end;
then (a 'imp' b) . x1 = TRUE by A3;
then A5: ('not' (a . x1)) 'or' (b . x1) = TRUE by BVFUNC_1:def 8;
(a '&' b) . x1 = (a . x1) '&' (b . x1) by MARGREL1:def 20
.= TRUE '&' TRUE by
.= TRUE ;
then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by ;
hence (Ex ((a '&' b),PA,G)) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum