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 holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u
let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Function of Y,BOOLEAN
for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u
let a, u be Function of 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 12 :: thesis: ( not (All ((a '&' u),PA,G)) . z = TRUE or ((Ex (a,PA,G)) '&' u) . z = TRUE )
A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume A2: (All ((a '&' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE
A3: 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 A2, BVFUNC_1:def 16;
then (a . x) '&' (u . x) = TRUE by MARGREL1:def 20;
hence ( a . x = TRUE & u . x = TRUE ) by MARGREL1:12; :: thesis: verum

end;

A4: ((Ex (a,PA,G)) '&' u) . z = ((Ex (a,PA,G)) . z) '&' (u . z) by MARGREL1:def 20;

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

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 17;
hence ((Ex (a,PA,G)) '&' u) . z = TRUE '&' TRUE by A4, A5, EQREL_1:def 6
.= 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 EQREL_1:def 6;
hence ((Ex (a,PA,G)) '&' u) . z = TRUE by A3, A1; :: 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

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

end;

end;