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 ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u
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 ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u
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 ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u )
assume A1: u is_independent_of PA,G ; :: thesis: All ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u
A2: (Ex (a,PA,G)) 'imp' u '<' All ((a 'imp' u),PA,G)

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (a,PA,G)) 'imp' u) . z = TRUE or (All ((a 'imp' u),PA,G)) . z = TRUE )
A3: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume ((Ex (a,PA,G)) 'imp' u) . z = TRUE ; :: thesis: (All ((a 'imp' u),PA,G)) . z = TRUE
then A4: ('not' ((Ex (a,PA,G)) . z)) 'or' (u . z) = TRUE by BVFUNC_1:def 8;
A5: ( 'not' ((Ex (a,PA,G)) . z) = TRUE or 'not' ((Ex (a,PA,G)) . z) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( u . z = TRUE & (All ((a 'imp' u),PA,G)) . z = TRUE ) or ( 'not' ((Ex (a,PA,G)) . z) = TRUE & u . z = FALSE & (All ((a 'imp' u),PA,G)) . z = TRUE ) )

per cases ( u . z = TRUE or ( 'not' ((Ex (a,PA,G)) . z) = TRUE & u . z = FALSE ) ) by A4, A5, BINARITH:3, XBOOLEAN:def 3;

case A6: u . z = TRUE ; :: thesis: (All ((a 'imp' u),PA,G)) . z = TRUE

for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' u) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'imp' u) . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'imp' u) . x = TRUE
then ( (a 'imp' u) . x = ('not' (a . x)) 'or' (u . x) & u . x = u . z ) by A1, A3, BVFUNC_1:def 8, BVFUNC_1:def 15;
hence (a 'imp' u) . x = TRUE by A6, BINARITH:10; :: thesis: verum

end;

hence (All ((a 'imp' u),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;

case A7: ( 'not' ((Ex (a,PA,G)) . z) = TRUE & u . z = FALSE ) ; :: thesis: (All ((a 'imp' u),PA,G)) . z = TRUE

for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' u) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'imp' u) . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'imp' u) . x = TRUE
then a . x <> TRUE by A7, BVFUNC_1:def 17, MARGREL1:11;
then ( (a 'imp' u) . x = ('not' (a . x)) 'or' (u . x) & a . x = FALSE ) by BVFUNC_1:def 8, XBOOLEAN:def 3;
then (a 'imp' u) . x = TRUE 'or' (u . x) by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence (a 'imp' u) . x = TRUE ; :: thesis: verum

end;

hence (All ((a 'imp' u),PA,G)) . z = TRUE by BVFUNC_1:def 16; :: thesis: verum

end;

end;

end;

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

end;

All ((a 'imp' u),PA,G) '<' (Ex (a,PA,G)) 'imp' u

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'imp' u),PA,G)) . z = TRUE or ((Ex (a,PA,G)) 'imp' u) . z = TRUE )
assume A8: (All ((a 'imp' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'imp' u) . z = TRUE
A9: ((Ex (a,PA,G)) 'imp' u) . z = ('not' ((Ex (a,PA,G)) . z)) 'or' (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 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))) & 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 holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

suppose for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
u . x = TRUE ; :: thesis: ((Ex (a,PA,G)) 'imp' u) . z = TRUE

then ((Ex (a,PA,G)) 'imp' u) . z = ('not' ((Ex (a,PA,G)) . z)) 'or' TRUE by A9, EQREL_1:def 6
.= TRUE by BINARITH:10 ;
hence ((Ex (a,PA,G)) 'imp' u) . z = TRUE ; :: thesis: verum

end;

suppose A10: ( 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))) & a . x = TRUE ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' u) . z = TRUE

then consider x1 being Element of Y such that
A11: x1 in EqClass (z,(CompF (PA,G))) and
A12: u . x1 <> TRUE ;
consider x2 being Element of Y such that
A13: x2 in EqClass (z,(CompF (PA,G))) and
A14: a . x2 = TRUE by A10;
A15: u . x1 = u . x2 by A1, A11, A13, BVFUNC_1:def 15;
(a 'imp' u) . x2 = ('not' (a . x2)) 'or' (u . x2) by BVFUNC_1:def 8
.= ('not' TRUE) 'or' FALSE by A12, A14, A15, XBOOLEAN:def 3
.= FALSE 'or' FALSE by MARGREL1:11
.= FALSE ;
hence ((Ex (a,PA,G)) 'imp' u) . z = TRUE by A8, A13, BVFUNC_1:def 16; :: thesis: verum

end;

suppose ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' u) . z = TRUE

then ((Ex (a,PA,G)) 'imp' u) . z = ('not' FALSE) 'or' (u . z) by A9, BVFUNC_1:def 17
.= TRUE 'or' (u . z) by MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ((Ex (a,PA,G)) 'imp' u) . z = TRUE ; :: thesis: verum

end;

end;

end;

hence All ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u by A2, BVFUNC_1:15; :: thesis: verum