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 PA,G holds
Ex u,PA,G '<' Ex 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 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 by BVFUNC_2:def 8;
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 8;
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 11;
A4: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

now

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 20;
then for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not u . x = TRUE ) by A1, A4, BVFUNC_1:def 18;
then (B_SUP u,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
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 14;
hence Ex u,PA,G '<' Ex u,PB,G by BVFUNC_1:19; :: thesis: verum