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

proof

let z be Element of Y; :: thesis: ((All (u,PA,G)) 'imp' (All (u,PB,G))) . z = TRUE
A2: ( z in EqClass (z,(CompF (PB,G))) & EqClass (z,(CompF (PB,G))) in CompF (PB,G) ) by EQREL_1:def 6;
A3: ((All (u,PA,G)) 'imp' (All (u,PB,G))) . z = ('not' ((All (u,PA,G)) . z)) 'or' ((All (u,PB,G)) . z) by BVFUNC_1:def 8;
A4: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

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

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

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

hence ((All (u,PA,G)) 'imp' (All (u,PB,G))) . z = TRUE by A3; :: thesis: verum

end;

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

then u . z = TRUE by A4, BVFUNC_1:def 16;
then for x being Element of Y st x in EqClass (z,(CompF (PB,G))) holds
u . x = TRUE by A1, A2;
then (All (u,PB,G)) . z = TRUE by BVFUNC_1:def 16;
hence ((All (u,PA,G)) 'imp' (All (u,PB,G))) . z = TRUE by A3; :: thesis: verum

end;

end;

end;

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

end;

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