let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs 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 Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds a '&' b '<' (Ex a,PA,G) '&' (Ex b,PA,G)
let a, b be Element of Funcs 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)
A1: Ex a,PA,G = B_SUP a,(CompF PA,G) by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (a '&' b) . z = TRUE or ((Ex a,PA,G) '&' (Ex b,PA,G)) . z = TRUE )
assume A2: (a '&' b) . z = TRUE ; :: thesis: ((Ex a,PA,G) '&' (Ex b,PA,G)) . z = TRUE
(a '&' b) . z = (a . z) '&' (b . z) by MARGREL1:def 21;
then A3: ( a . z = TRUE & b . z = TRUE ) by A2, MARGREL1:45;
A4: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
then (B_SUP b,(CompF PA,G)) . z = TRUE by A3, BVFUNC_1:def 20;
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 21
.= TRUE '&' TRUE by A1, A3, A4, A5, BVFUNC_1:def 20
.= TRUE ; :: thesis: verum