let Y be non empty set ; :: thesis: for u being Element of Funcs 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 Element of Funcs 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 by BVFUNC_2:def 8;
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 8;
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 11;
A4: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;

now

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 19;
then for x being Element of Y st x in EqClass z,(CompF PB,G) holds
u . x = TRUE by A1, A2, BVFUNC_1:def 18;
then (All u,PB,G) . z = TRUE by BVFUNC_1:def 19;
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 14;
hence All u,PA,G '<' All u,PB,G by BVFUNC_1:19; :: thesis: verum