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

proof

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

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

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

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

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

end;

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

then u . z <> TRUE by A2, BVFUNC_1:def 17;
then for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not u . x = TRUE ) by A1, A4;
then (B_SUP (u,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;
hence ((Ex (u,PA,G)) 'imp' (Ex (u,PB,G))) . z = TRUE by A3; :: thesis: verum

end;

end;

end;

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

end;

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