let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All (b,PA,G)) '<' Ex ((a '&' b),PA,G)
let G be Subset of (PARTITIONS Y); for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All (b,PA,G)) '<' Ex ((a '&' b),PA,G)
let a, b be Function of Y,BOOLEAN; for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All (b,PA,G)) '<' Ex ((a '&' b),PA,G)
let PA be a_partition of Y; (Ex (a,PA,G)) '&' (All (b,PA,G)) '<' Ex ((a '&' b),PA,G)
let z be Element of Y; BVFUNC_1:def 12 ( not ((Ex (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE or (Ex ((a '&' b),PA,G)) . z = TRUE )
assume
((Ex (a,PA,G)) '&' (All (b,PA,G))) . z = TRUE
; (Ex ((a '&' b),PA,G)) . z = TRUE
then A1:
((Ex (a,PA,G)) . z) '&' ((All (b,PA,G)) . z) = TRUE
by MARGREL1:def 20;
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 =
(a . x1) '&' (b . x1)
by MARGREL1:def 20
.=
TRUE '&' TRUE
by A3, A4, A2
.=
TRUE
;
then
(B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE
by A3, BVFUNC_1:def 17;
hence
(Ex ((a '&' b),PA,G)) . z = TRUE
by BVFUNC_2:def 10; verum