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

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

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

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

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

then consider x1 being Element of Y such that

A1: x1 in EqClass (z,(CompF (PA,G))) and

A2: (a '&' b) . x1 = TRUE by BVFUNC_1:def 17;

A3: (a . x1) '&' (b . x1) = TRUE by A2, MARGREL1:def 20;

then A4: b . x1 = TRUE by MARGREL1:12;

a . x1 = TRUE by A3, MARGREL1:12;

then A5: (Ex (a,PA,G)) . z = TRUE by A1, BVFUNC_1:def 17;

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

.= TRUE '&' TRUE by A1, A4, A5, BVFUNC_1:def 17

.= TRUE ;

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

for G being Subset of (PARTITIONS Y)

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

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

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

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

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

then consider x1 being Element of Y such that

A1: x1 in EqClass (z,(CompF (PA,G))) and

A2: (a '&' b) . x1 = TRUE by BVFUNC_1:def 17;

A3: (a . x1) '&' (b . x1) = TRUE by A2, MARGREL1:def 20;

then A4: b . x1 = TRUE by MARGREL1:12;

a . x1 = TRUE by A3, MARGREL1:12;

then A5: (Ex (a,PA,G)) . z = TRUE by A1, BVFUNC_1:def 17;

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

.= TRUE '&' TRUE by A1, A4, A5, BVFUNC_1:def 17

.= TRUE ;

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