let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)

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 (PARTITIONS Y); :: 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;

A3: x1 in EqClass (z,(CompF (PA,G))) and

A4: a . x1 = TRUE ;

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 A4, A5, A1, BINARITH:3

.= TRUE ;

then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by A3, BVFUNC_1:def 17;

hence (Ex ((a '&' b),PA,G)) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum

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 (PARTITIONS Y); :: 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 )

then consider x1 being Element of Y such that ( 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;( 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

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

then
(a 'imp' b) . x1 = TRUE
by A3;(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;( 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

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 A4, A5, A1, BINARITH:3

.= TRUE ;

then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by A3, BVFUNC_1:def 17;

hence (Ex ((a '&' b),PA,G)) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum