let Y be non empty set ; :: thesis: for u being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st u is_independent_of PA,G holds
u '<' All (u,PA,G)
let u be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st u is_independent_of PA,G holds
u '<' All (u,PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds
u '<' All (u,PA,G)
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies u '<' All (u,PA,G) )
assume u is_independent_of PA,G ; :: thesis: u '<' All (u,PA,G)
then A1: u is_dependent_of CompF (PA,G) ;
for z being Element of Y holds (u 'imp' (All (u,PA,G))) . z = TRUE

proof

let z be Element of Y; :: thesis: (u 'imp' (All (u,PA,G))) . z = TRUE
A2: (u 'imp' (All (u,PA,G))) . z = ('not' (u . z)) 'or' ((All (u,PA,G)) . z) by BVFUNC_1:def 8;
A3: ( z in EqClass (z,(CompF (PA,G))) & EqClass (z,(CompF (PA,G))) in CompF (PA,G) ) by EQREL_1:def 6;

now :: thesis: ( ( u . z = FALSE & (u 'imp' (All (u,PA,G))) . z = TRUE ) or ( u . z = TRUE & (u 'imp' (All (u,PA,G))) . z = TRUE ) )

per cases ( u . z = FALSE or u . z = TRUE ) by XBOOLEAN:def 3;

case u . z = FALSE ; :: thesis: (u 'imp' (All (u,PA,G))) . z = TRUE

hence (u 'imp' (All (u,PA,G))) . z = TRUE by A2; :: thesis: verum

end;

case u . z = TRUE ; :: thesis: (u 'imp' (All (u,PA,G))) . z = TRUE

then for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
u . x = TRUE by A1, A3;
then (All (u,PA,G)) . z = TRUE by BVFUNC_1:def 16;
hence (u 'imp' (All (u,PA,G))) . z = TRUE by A2; :: thesis: verum

end;

end;

end;

hence (u 'imp' (All (u,PA,G))) . z = TRUE ; :: thesis: verum

end;

then u 'imp' (All (u,PA,G)) = I_el Y by BVFUNC_1:def 11;
hence u '<' All (u,PA,G) by BVFUNC_1:16; :: thesis: verum