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

hence Ex (u,PA,G) '<' Ex (u,PB,G) by BVFUNC_1:16; :: thesis: verum

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

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

end;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 ) )end;

hence
((Ex (u,PA,G)) 'imp' (Ex (u,PB,G))) . z = TRUE
; :: thesis: verumper cases
( (Ex (u,PB,G)) . z = TRUE or (Ex (u,PB,G)) . z = FALSE )
by XBOOLEAN:def 3;

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;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

hence Ex (u,PA,G) '<' Ex (u,PB,G) by BVFUNC_1:16; :: thesis: verum