let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds B_SUP a,PA is_dependent_of PA
let a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds B_SUP a,PA is_dependent_of PA
let PA be a_partition of Y; :: thesis: B_SUP a,PA is_dependent_of PA
for F being set st F in PA holds
for x1, x2 being set st x1 in F & x2 in F holds
(B_SUP a,PA) . x1 = (B_SUP a,PA) . x2

proof

let F be set ; :: thesis: ( F in PA implies for x1, x2 being set st x1 in F & x2 in F holds
(B_SUP a,PA) . x1 = (B_SUP a,PA) . x2 )
assume A1: F in PA ; :: thesis: for x1, x2 being set st x1 in F & x2 in F holds
(B_SUP a,PA) . x1 = (B_SUP a,PA) . x2
let x1, x2 be set ; :: thesis: ( x1 in F & x2 in F implies (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2 )
assume A2: ( x1 in F & x2 in F ) ; :: thesis: (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2
then reconsider x1 = x1, x2 = x2 as Element of Y by A1;
A3: x1 in EqClass x1,PA by EQREL_1:def 8;
( EqClass x1,PA = F or EqClass x1,PA misses F ) by A1, EQREL_1:def 6;
then A4: EqClass x2,PA = EqClass x1,PA by A2, A3, EQREL_1:def 8, XBOOLE_0:3;

per cases ( ex x being Element of Y st
( x in EqClass x1,PA & a . x = TRUE ) or for x being Element of Y holds
( not x in EqClass x1,PA or not a . x = TRUE ) ) ;

suppose A5: ex x being Element of Y st
( x in EqClass x1,PA & a . x = TRUE ) ; :: thesis: (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2

then (B_SUP a,PA) . x1 = TRUE by Def20;
hence (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2 by A4, A5, Def20; :: thesis: verum

end;

suppose A6: for x being Element of Y holds
( not x in EqClass x1,PA or not a . x = TRUE ) ; :: thesis: (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2

then (B_SUP a,PA) . x1 = FALSE by Def20;
hence (B_SUP a,PA) . x1 = (B_SUP a,PA) . x2 by A4, A6, Def20; :: thesis: verum

end;

end;

end;

hence B_SUP a,PA is_dependent_of PA by Def18; :: thesis: verum