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 (All (a,PA,G)) '&' (All (b,PA,G)) '<' a '&' b

let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds (All (a,PA,G)) '&' (All (b,PA,G)) '<' a '&' b

let a, b be Function of 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 12 :: thesis: ( not ((All (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE or (a '&' b) . z = TRUE )

A1: ((All (a,PA,G)) '&' (All (b,PA,G))) . z = ((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z) by MARGREL1:def 20;

assume A2: ((All (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE ; :: thesis: (a '&' b) . z = TRUE

thus (a '&' b) . z = (a . z) '&' (b . z) by MARGREL1:def 20

.= TRUE '&' TRUE by A4, A3, A5

.= TRUE ; :: thesis: verum

for a, b being Function of 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 Function of Y,BOOLEAN

for PA being a_partition of Y holds (All (a,PA,G)) '&' (All (b,PA,G)) '<' a '&' b

let a, b be Function of 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 12 :: thesis: ( not ((All (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE or (a '&' b) . z = TRUE )

A1: ((All (a,PA,G)) '&' (All (b,PA,G))) . z = ((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z) by MARGREL1:def 20;

assume A2: ((All (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE ; :: thesis: (a '&' b) . z = TRUE

A3: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

a . x = TRUE

A4:
z in EqClass (z,(CompF (PA,G)))
by EQREL_1:def 6;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 (B_INF (a,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;

then (All (a,PA,G)) . z = FALSE by BVFUNC_2:def 9;

hence contradiction by A2, A1, MARGREL1:12; :: thesis: verum

end;( 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 16;

then (All (a,PA,G)) . z = FALSE by BVFUNC_2:def 9;

hence contradiction by A2, A1, MARGREL1:12; :: thesis: verum

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

b . x = TRUE

then A5:
b . z = TRUE
by A4;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 (B_INF (b,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;

then (All (b,PA,G)) . z = FALSE by BVFUNC_2:def 9;

hence contradiction by A2, A1, MARGREL1:12; :: thesis: verum

end;( 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 16;

then (All (b,PA,G)) . z = FALSE by BVFUNC_2:def 9;

hence contradiction by A2, A1, MARGREL1:12; :: thesis: verum

thus (a '&' b) . z = (a . z) '&' (b . z) by MARGREL1:def 20

.= TRUE '&' TRUE by A4, A3, A5

.= TRUE ; :: thesis: verum