let Y be non empty set ; :: thesis: for a, b being Element of Funcs 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 Element of Funcs 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 15 :: 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 20;
A3: (a . x1) '&' (b . x1) = TRUE by A2, MARGREL1:def 21;
then A4: b . x1 = TRUE by MARGREL1:45;
a . x1 = TRUE by A3, MARGREL1:45;
then A5: (Ex a,PA,G) . z = TRUE by A1, BVFUNC_1:def 20;
((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, A4, A5, BVFUNC_1:def 20
.= TRUE ;
hence ((Ex a,PA,G) '&' (Ex b,PA,G)) . z = TRUE ; :: thesis: verum