let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN

for G being Subset of (PARTITIONS Y)

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

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

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

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

let PA be a_partition of Y; :: thesis: All ((a '&' b),PA,G) '<' a '&' (All (b,PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a '&' b),PA,G)) . z = TRUE or (a '&' (All (b,PA,G))) . z = TRUE )

assume A1: (All ((a '&' b),PA,G)) . z = TRUE ; :: thesis: (a '&' (All (b,PA,G))) . z = TRUE

z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

then a . z = TRUE by A2;

then (a '&' (All (b,PA,G))) . z = TRUE '&' TRUE by A9, MARGREL1:def 20

.= TRUE ;

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

for G being Subset of (PARTITIONS Y)

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

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

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

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

let PA be a_partition of Y; :: thesis: All ((a '&' b),PA,G) '<' a '&' (All (b,PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a '&' b),PA,G)) . z = TRUE or (a '&' (All (b,PA,G))) . z = TRUE )

assume A1: (All ((a '&' b),PA,G)) . z = TRUE ; :: thesis: (a '&' (All (b,PA,G))) . z = TRUE

A2: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

a . x = TRUE

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 consider x1 being Element of Y such that

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

A4: a . x1 <> TRUE ;

(a '&' b) . x1 = TRUE by A1, A3, BVFUNC_1:def 16;

then A5: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;

a . x1 = FALSE by A4, XBOOLEAN:def 3;

hence contradiction by A5; :: thesis: verum

end;( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) ; :: thesis: contradiction

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 = TRUE by A1, A3, BVFUNC_1:def 16;

then A5: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;

a . x1 = FALSE by A4, XBOOLEAN:def 3;

hence contradiction by A5; :: thesis: verum

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

b . x = TRUE

then A9:
(B_INF (b,(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 16;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 consider x1 being Element of Y such that

A6: x1 in EqClass (z,(CompF (PA,G))) and

A7: b . x1 <> TRUE ;

(a '&' b) . x1 = TRUE by A1, A6, BVFUNC_1:def 16;

then A8: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;

b . x1 = FALSE by A7, XBOOLEAN:def 3;

hence contradiction by A8; :: thesis: verum

end;( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ; :: thesis: contradiction

then consider x1 being Element of Y such that

A6: x1 in EqClass (z,(CompF (PA,G))) and

A7: b . x1 <> TRUE ;

(a '&' b) . x1 = TRUE by A1, A6, BVFUNC_1:def 16;

then A8: (a . x1) '&' (b . x1) = TRUE by MARGREL1:def 20;

b . x1 = FALSE by A7, XBOOLEAN:def 3;

hence contradiction by A8; :: thesis: verum

z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

then a . z = TRUE by A2;

then (a '&' (All (b,PA,G))) . z = TRUE '&' TRUE by A9, MARGREL1:def 20

.= TRUE ;

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