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

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

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

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

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

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

then A3: ( Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) & a . z = TRUE ) by A1, BVFUNC_2:def 10, MARGREL1:12;

A4: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

b . z = TRUE by A2, A1, MARGREL1:12;

then (B_SUP (b,(CompF (PA,G)))) . z = TRUE by A4, BVFUNC_1:def 17;

then A5: (Ex (b,PA,G)) . z = TRUE by BVFUNC_2:def 10;

thus ((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 A3, A4, A5, BVFUNC_1:def 17

.= TRUE ; :: thesis: verum

for a, b being Function of Y,BOOLEAN

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

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

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

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

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

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

then A3: ( Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) & a . z = TRUE ) by A1, BVFUNC_2:def 10, MARGREL1:12;

A4: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

b . z = TRUE by A2, A1, MARGREL1:12;

then (B_SUP (b,(CompF (PA,G)))) . z = TRUE by A4, BVFUNC_1:def 17;

then A5: (Ex (b,PA,G)) . z = TRUE by BVFUNC_2:def 10;

thus ((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 A3, A4, A5, BVFUNC_1:def 17

.= TRUE ; :: thesis: verum