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 'eqv' a),PA,G '<' u 'eqv' (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 'eqv' a),PA,G '<' u 'eqv' (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 'eqv' a),PA,G '<' u 'eqv' (All a,PA,G)
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All (u 'eqv' a),PA,G '<' u 'eqv' (All a,PA,G) )
assume
u is_independent_of PA,G
; :: thesis: All (u 'eqv' a),PA,G '<' u 'eqv' (All a,PA,G)
then A1:
u is_dependent_of CompF PA,G
by Def8;
A2:
( 'not' FALSE = TRUE & 'not' TRUE = FALSE )
by MARGREL1:41;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (u 'eqv' a),PA,G) . z = TRUE or (u 'eqv' (All a,PA,G)) . z = TRUE )
assume A3:
(All (u 'eqv' a),PA,G) . z = TRUE
; :: thesis: (u 'eqv' (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;
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 ) & ( 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 u . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not a . x = TRUE ) ) )
;
suppose A5:
( ( 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 ) )
;
:: thesis: (u 'eqv' (All a,PA,G)) . z = TRUE then A6:
(All a,PA,G) . z = TRUE
by BVFUNC_1:def 19;
A7:
u . z = TRUE
by A4, A5;
(u 'eqv' (All a,PA,G)) . z =
'not' ((u . z) 'xor' ((All a,PA,G) . z))
by BVFUNC_1:def 12
.=
TRUE
by A2, A6, A7, MARGREL1:49
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
;
:: thesis: verum end; suppose A8:
( ( 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 'eqv' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A9:
(
x1 in EqClass z,
(CompF PA,G) &
a . x1 <> TRUE )
;
A10:
u . x1 = TRUE
by A8, A9;
A11:
a . x1 = FALSE
by A9, XBOOLEAN:def 3;
(u 'eqv' a) . x1 =
'not' ((u . x1) 'xor' (a . x1))
by BVFUNC_1:def 12
.=
'not' (FALSE 'or' TRUE )
by A2, A10, A11
.=
'not' TRUE
by BINARITH:7
.=
FALSE
by MARGREL1:41
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
by A3, A9, BVFUNC_1:def 19;
:: thesis: verum end; suppose A12:
( ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
u . x = TRUE ) & ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE ) )
;
:: thesis: (u 'eqv' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A13:
(
x1 in EqClass z,
(CompF PA,G) &
u . x1 <> TRUE )
;
A14:
u . x1 = FALSE
by A13, XBOOLEAN:def 3;
A15:
a . x1 = TRUE
by A12, A13;
(u 'eqv' a) . x1 =
'not' ((u . x1) 'xor' (a . x1))
by BVFUNC_1:def 12
.=
'not' (TRUE 'or' FALSE )
by A2, A14, A15
.=
'not' TRUE
by BINARITH:7
.=
FALSE
by MARGREL1:41
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
by A3, A13, BVFUNC_1:def 19;
:: thesis: verum end; suppose A16:
( ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
u . x = TRUE ) & ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE ) )
;
:: thesis: (u 'eqv' (All a,PA,G)) . z = TRUE then A17:
(All a,PA,G) . z = FALSE
by BVFUNC_1:def 19;
consider x1 being
Element of
Y such that A18:
(
x1 in EqClass z,
(CompF PA,G) &
u . x1 <> TRUE )
by A16;
u . x1 = u . z
by A1, A4, A18, BVFUNC_1:def 18;
then A19:
u . z = FALSE
by A18, XBOOLEAN:def 3;
(u 'eqv' (All a,PA,G)) . z =
'not' ((u . z) 'xor' ((All a,PA,G) . z))
by BVFUNC_1:def 12
.=
TRUE
by A2, A17, A19, MARGREL1:49
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
;
:: thesis: verum end;