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 ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u
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 ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u
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 ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u )
assume u is_independent_of PA,G ; :: thesis: All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u
then A1: u is_dependent_of CompF (PA,G) by Def8;
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'or' u),PA,G)) . z = TRUE or ((Ex (a,PA,G)) 'or' u) . z = TRUE )
A2: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume A3: (All ((a 'or' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE
A4: for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or a . x = TRUE or u . x = TRUE )

proof

let x be Element of Y; :: thesis: ( not x in EqClass (z,(CompF (PA,G))) or a . x = TRUE or u . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ( a . x = TRUE or u . x = TRUE )
then (a 'or' u) . x = TRUE by A3, BVFUNC_1:def 16;
then A5: (a . x) 'or' (u . x) = TRUE by BVFUNC_1:def 4;
( u . x = TRUE or u . x = FALSE ) by XBOOLEAN:def 3;
hence ( a . x = TRUE or u . x = TRUE ) by A5, BINARITH:3; :: thesis: verum

end;

A6: ((Ex (a,PA,G)) 'or' u) . z = ((Ex (a,PA,G)) . z) 'or' (u . z) by BVFUNC_1:def 4;

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)) 'or' u) . z = TRUE

then ((Ex (a,PA,G)) 'or' u) . z = ((Ex (a,PA,G)) . z) 'or' TRUE by A2, A6
.= TRUE by BINARITH:10 ;
hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

end;

suppose ( 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)) 'or' u) . z = TRUE

then ((Ex (a,PA,G)) 'or' u) . z = TRUE 'or' (u . z) by A6, BVFUNC_1:def 17
.= TRUE by BINARITH:10 ;
hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

end;

suppose A7: ( 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)) 'or' u) . z = TRUE

A8: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
then A9: a . z <> TRUE by A7;
consider x1 being Element of Y such that
A10: x1 in EqClass (z,(CompF (PA,G))) and
A11: u . x1 <> TRUE by A7;
u . x1 = u . z by A1, A8, A10, BVFUNC_1:def 15;
hence ((Ex (a,PA,G)) 'or' u) . z = TRUE by A4, A8, A9, A11; :: thesis: verum

end;

end;