let Y be non empty set ; :: thesis: 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
All ((u 'eqv' a),PA,G) '<' u 'eqv' (All (a,PA,G))
let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Function of 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 a, u be Function of 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)) )
A1: 'not' FALSE = TRUE by MARGREL1:11;
assume A2: u is_independent_of PA,G ; :: thesis: All ((u 'eqv' a),PA,G) '<' u 'eqv' (All (a,PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: 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))) by EQREL_1:def 6;

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 ) ) ; :: thesis: (u 'eqv' (All (a,PA,G))) . z = TRUE

then A5: ( (All (a,PA,G)) . z = TRUE & u . z = TRUE ) by BVFUNC_1:def 16, EQREL_1:def 6;
(u 'eqv' (All (a,PA,G))) . z = 'not' ((u . z) 'xor' ((All (a,PA,G)) . z)) by BVFUNC_1:def 9
.= TRUE by A1, A5, MARGREL1:13 ;
hence (u 'eqv' (All (a,PA,G))) . z = TRUE ; :: thesis: 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 ) ) ; :: thesis: (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 9
.= 'not' (FALSE 'or' TRUE) by A1, A9, A10
.= FALSE by BINARITH:3, MARGREL1:11 ;
hence (u 'eqv' (All (a,PA,G))) . z = TRUE by A3, A7, BVFUNC_1:def 16; :: thesis: 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 ) ) ; :: thesis: (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 9
.= 'not' (TRUE 'or' FALSE) by A1, A15, A14
.= FALSE by BINARITH:3, MARGREL1:11 ;
hence (u 'eqv' (All (a,PA,G))) . z = TRUE by A3, A12, BVFUNC_1:def 16; :: 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 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 15;
then A19: u . z = FALSE by A18, XBOOLEAN:def 3;
A20: (All (a,PA,G)) . z = FALSE by A16, BVFUNC_1:def 16;
(u 'eqv' (All (a,PA,G))) . z = 'not' ((u . z) 'xor' ((All (a,PA,G)) . z)) by BVFUNC_1:def 9
.= TRUE by A20, A19, MARGREL1:11, MARGREL1:13 ;
hence (u 'eqv' (All (a,PA,G))) . z = TRUE ; :: thesis: verum

end;

end;