let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
All (u 'xor' a),PA,G '<' u 'xor' (All a,PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
All (u 'xor' a),PA,G '<' u 'xor' (All a,PA,G)
let u, a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds
All (u 'xor' a),PA,G '<' u 'xor' (All a,PA,G)
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All (u 'xor' a),PA,G '<' u 'xor' (All a,PA,G) )
A1:
( 'not' FALSE = TRUE & 'not' TRUE = FALSE )
by MARGREL1:41;
assume
u is_independent_of PA,G
; :: thesis: All (u 'xor' a),PA,G '<' u 'xor' (All a,PA,G)
then A2:
u is_dependent_of CompF PA,G
by Def8;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (u 'xor' a),PA,G) . z = TRUE or (u 'xor' (All a,PA,G)) . z = TRUE )
assume A3:
(All (u 'xor' a),PA,G) . z = TRUE
; :: thesis: (u 'xor' (All a,PA,G)) . z = TRUE
A4:
( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G )
by EQREL_1:def 8;
A5:
(u 'xor' (All a,PA,G)) . z = ((All a,PA,G) . z) 'xor' (u . z)
by BVFUNC_1:def 8;
per cases
( ( ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
u . x = TRUE ) & ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
a . x = TRUE ) ) or ( ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
u . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) ) or ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not u . x = TRUE ) )
;
suppose A10:
( ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
u . x = TRUE ) & ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE ) )
;
:: thesis: (u 'xor' (All a,PA,G)) . z = TRUE then A11:
(All a,PA,G) . z = FALSE
by BVFUNC_1:def 19;
consider x1 being
Element of
Y such that A12:
(
x1 in EqClass z,
(CompF PA,G) &
a . x1 <> TRUE )
by A10;
A13:
u . x1 = TRUE
by A10, A12;
u . z = u . x1
by A2, A4, A12, BVFUNC_1:def 18;
then (u 'xor' (All a,PA,G)) . z =
TRUE 'or' FALSE
by A1, A5, A11, A13
.=
TRUE
by BINARITH:7
;
hence
(u 'xor' (All a,PA,G)) . z = TRUE
;
:: thesis: verum end; suppose
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
u . x = TRUE )
;
:: thesis: (u 'xor' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A14:
(
x1 in EqClass z,
(CompF PA,G) &
u . x1 <> TRUE )
;
now per cases
( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
a . x = TRUE or ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) )
;
suppose
for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE
;
:: thesis: (u 'xor' (All a,PA,G)) . z = TRUE then A15:
(All a,PA,G) . z = TRUE
by BVFUNC_1:def 19;
u . z = u . x1
by A2, A4, A14, BVFUNC_1:def 18;
then
u . z = FALSE
by A14, XBOOLEAN:def 3;
then (u 'xor' (All a,PA,G)) . z =
FALSE 'or' TRUE
by A1, A5, A15
.=
TRUE
by BINARITH:7
;
hence
(u 'xor' (All a,PA,G)) . z = TRUE
;
:: thesis: verum end; suppose
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE )
;
:: thesis: (u 'xor' (All a,PA,G)) . z = TRUE then consider x2 being
Element of
Y such that A16:
(
x2 in EqClass z,
(CompF PA,G) &
a . x2 <> TRUE )
;
u . x1 = u . x2
by A2, A14, A16, BVFUNC_1:def 18;
then A17:
u . x2 = FALSE
by A14, XBOOLEAN:def 3;
A18:
a . x2 = FALSE
by A16, XBOOLEAN:def 3;
(u 'xor' a) . x2 =
(a . x2) 'xor' (u . x2)
by BVFUNC_1:def 8
.=
FALSE
by A17, A18, MARGREL1:45
;
hence
(u 'xor' (All a,PA,G)) . z = TRUE
by A3, A16, BVFUNC_1:def 19;
:: thesis: verum end; hence
(u 'xor' (All a,PA,G)) . z = TRUE
;
:: thesis: verum end;