let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, u being Function of Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G))
let G be Subset of (PARTITIONS Y); for a, u being Function of Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G))
let a, u be Function of Y,BOOLEAN; for PA being a_partition of Y st u is_independent_of PA,G holds
Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G))
let PA be a_partition of Y; ( u is_independent_of PA,G implies Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G)) )
assume
u is_independent_of PA,G
; Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G))
then A1:
u is_dependent_of CompF (PA,G)
by BVFUNC_2:def 8;
let z be Element of Y; BVFUNC_1:def 12 ( not (Ex ((u 'imp' a),PA,G)) . z = TRUE or (u 'imp' (Ex (a,PA,G))) . z = TRUE )
A2:
( z in EqClass (z,(CompF (PA,G))) & EqClass (z,(CompF (PA,G))) in CompF (PA,G) )
by EQREL_1:def 6;
assume A3:
(Ex ((u 'imp' a),PA,G)) . z = TRUE
; (u 'imp' (Ex (a,PA,G))) . z = TRUE
then consider x1 being Element of Y such that
A4:
x1 in EqClass (z,(CompF (PA,G)))
and
A5:
(u 'imp' a) . x1 = TRUE
;
A6:
('not' (u . x1)) 'or' (a . x1) = TRUE
by A5, BVFUNC_1:def 8;
A7:
( 'not' (u . x1) = TRUE or 'not' (u . x1) = FALSE )
by XBOOLEAN:def 3;