let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, u being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds All (a '&' u),PA,G '<' (Ex a,PA,G) '&' u
let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds All (a '&' u),PA,G '<' (Ex a,PA,G) '&' u
let a, u be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds All (a '&' u),PA,G '<' (Ex a,PA,G) '&' u
let PA be a_partition of Y; :: thesis: All (a '&' u),PA,G '<' (Ex a,PA,G) '&' u
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (a '&' u),PA,G) . z = TRUE or ((Ex a,PA,G) '&' u) . z = TRUE )
assume A1: (All (a '&' u),PA,G) . z = TRUE ; :: thesis: ((Ex a,PA,G) '&' u) . z = TRUE
A2: for x being Element of Y st x in EqClass z,(CompF PA,G) holds
( a . x = TRUE & u . x = TRUE )

proof

let x be Element of Y; :: thesis: ( x in EqClass z,(CompF PA,G) implies ( a . x = TRUE & u . x = TRUE ) )
assume x in EqClass z,(CompF PA,G) ; :: thesis: ( a . x = TRUE & u . x = TRUE )
then (a '&' u) . x = TRUE by A1, BVFUNC_1:def 19;
then (a . x) '&' (u . x) = TRUE by MARGREL1:def 21;
hence ( a . x = TRUE & u . x = TRUE ) by MARGREL1:45; :: thesis: verum

end;

A3: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
A4: ((Ex a,PA,G) '&' u) . z = ((Ex a,PA,G) . z) '&' (u . z) by MARGREL1:def 21;

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 ) ) ;

suppose A5: for x being Element of Y st x in EqClass z,(CompF PA,G) holds
u . x = TRUE ; :: thesis: ((Ex a,PA,G) '&' u) . z = TRUE

now

per cases ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & a . x = TRUE ) or for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not a . x = TRUE ) ) ;

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

then (Ex a,PA,G) . z = TRUE by BVFUNC_1:def 20;
hence ((Ex a,PA,G) '&' u) . z = TRUE '&' TRUE by A3, A4, A5
.= TRUE ;
:: thesis: verum

end;

suppose 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) '&' u) . z = TRUE

then a . z <> TRUE by A3;
hence ((Ex a,PA,G) '&' u) . z = TRUE by A2, A3; :: thesis: verum

end;

end;

end;

hence ((Ex a,PA,G) '&' 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 ) ; :: thesis: ((Ex a,PA,G) '&' u) . z = TRUE

then consider x1 being Element of Y such that
A6: ( x1 in EqClass z,(CompF PA,G) & u . x1 <> TRUE ) ;
thus ((Ex a,PA,G) '&' u) . z = TRUE by A2, A6; :: thesis: verum

end;

end;