let Y be non empty set ; 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); 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 ; 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; ( u is_independent_of PA,G implies All (u 'eqv' a),PA,G '<' u 'eqv' (All a,PA,G) )
A1:
'not' FALSE = TRUE
by MARGREL1:41;
assume
u is_independent_of PA,G
; All (u 'eqv' a),PA,G '<' u 'eqv' (All a,PA,G)
then A2:
u is_dependent_of CompF PA,G
by Def8;
let z be Element of Y; BVFUNC_1:def 15 ( 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
; (u 'eqv' (All a,PA,G)) . z = TRUE
A4:
z in EqClass z,(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
( ( 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 ) )
;
(u 'eqv' (All a,PA,G)) . z = TRUE then A5:
(
(All a,PA,G) . z = TRUE &
u . z = TRUE )
by A4, BVFUNC_1:def 19;
(u 'eqv' (All a,PA,G)) . z =
'not' ((u . z) 'xor' ((All a,PA,G) . z))
by BVFUNC_1:def 12
.=
TRUE
by A1, A5, MARGREL1:49
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
;
verum end; suppose A6:
( ( 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 ) )
;
(u 'eqv' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A7:
x1 in EqClass z,
(CompF PA,G)
and A8:
a . x1 <> TRUE
;
A9:
u . x1 = TRUE
by A6, A7;
A10:
a . x1 = FALSE
by A8, XBOOLEAN:def 3;
(u 'eqv' a) . x1 =
'not' ((u . x1) 'xor' (a . x1))
by BVFUNC_1:def 12
.=
'not' (FALSE 'or' TRUE )
by A1, A9, A10
.=
'not' TRUE
by BINARITH:7
.=
FALSE
by MARGREL1:41
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
by A3, A7, BVFUNC_1:def 19;
verum end; suppose A11:
( 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 ) )
;
(u 'eqv' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A12:
x1 in EqClass z,
(CompF PA,G)
and A13:
u . x1 <> TRUE
;
A14:
a . x1 = TRUE
by A11, A12;
A15:
u . x1 = FALSE
by A13, XBOOLEAN:def 3;
(u 'eqv' a) . x1 =
'not' ((u . x1) 'xor' (a . x1))
by BVFUNC_1:def 12
.=
'not' (TRUE 'or' FALSE )
by A1, A15, A14
.=
'not' TRUE
by BINARITH:7
.=
FALSE
by MARGREL1:41
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
by A3, A12, BVFUNC_1:def 19;
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 ) )
;
(u 'eqv' (All a,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A17:
x1 in EqClass z,
(CompF PA,G)
and A18:
u . x1 <> TRUE
;
u . x1 = u . z
by A2, A4, A17, BVFUNC_1:def 18;
then A19:
u . z = FALSE
by A18, XBOOLEAN:def 3;
A20:
(All a,PA,G) . z = FALSE
by A16, BVFUNC_1:def 19;
(u 'eqv' (All a,PA,G)) . z =
'not' ((u . z) 'xor' ((All a,PA,G) . z))
by BVFUNC_1:def 12
.=
TRUE
by A1, A20, A19, MARGREL1:49
;
hence
(u 'eqv' (All a,PA,G)) . z = TRUE
;
verum end;