let Y be non empty set ; 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; 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); 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; ( u is_independent_of PA,G implies Ex (u,PA,G) '<' Ex (u,PB,G) )
assume
u is_independent_of PA,G
; 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;
((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 ( ( (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 = FALSE
;
((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;
verum end; end; end;
hence
((Ex (u,PA,G)) 'imp' (Ex (u,PB,G))) . z = TRUE
;
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; verum