let Y be non empty set ; :: thesis: for u being Element of Funcs 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 Element of Funcs 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 by BVFUNC_2:def 8;
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 11;
A3: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

now

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, BVFUNC_1:def 18;
then (All u,PA,G) . z = TRUE by BVFUNC_1:def 19;
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 14;
hence u '<' All u,PA,G by BVFUNC_1:19; :: thesis: verum