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 15 :: thesis: ( not (All (a 'or' u),PA,G) . z = TRUE or ((Ex a,PA,G) 'or' u) . z = TRUE )
assume A2: (All (a 'or' u),PA,G) . z = TRUE ; :: thesis: ((Ex a,PA,G) 'or' u) . z = TRUE
A3: 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 A2, BVFUNC_1:def 19;
then A4: (a . x) 'or' (u . x) = TRUE by BVFUNC_1:def 7;
( u . x = TRUE or u . x = FALSE ) by XBOOLEAN:def 3;
hence ( a . x = TRUE or u . x = TRUE ) by A4, BINARITH:7; :: thesis: verum

end;

A5: z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
A6: ((Ex a,PA,G) 'or' u) . z = ((Ex a,PA,G) . z) 'or' (u . z) by BVFUNC_1:def 7;

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 A5, A6
.= TRUE by BINARITH:19 ;
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 20
.= TRUE by BINARITH:19 ;
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) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
then A9: a . z <> TRUE by A7;
consider x1 being Element of Y such that
A10: ( x1 in EqClass z,(CompF PA,G) & u . x1 <> TRUE ) by A7;
u . x1 = u . z by A1, A8, A10, BVFUNC_1:def 18;
hence ((Ex a,PA,G) 'or' u) . z = TRUE by A3, A8, A9, A10; :: thesis: verum

end;

end;