let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds B_INF a,PA is_dependent_of PA
let a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds B_INF a,PA is_dependent_of PA
let PA be a_partition of Y; :: thesis: B_INF 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_INF a,PA) . x1 = (B_INF 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_INF a,PA) . x1 = (B_INF a,PA) . x2 )
assume A1: F in PA ; :: thesis: for x1, x2 being set st x1 in F & x2 in F holds
(B_INF a,PA) . x1 = (B_INF a,PA) . x2
let x1, x2 be set ; :: thesis: ( x1 in F & x2 in F implies (B_INF a,PA) . x1 = (B_INF a,PA) . x2 )
assume that
A2: x1 in F and
A3: x2 in F ; :: thesis: (B_INF a,PA) . x1 = (B_INF a,PA) . x2
reconsider x1 = x1 as Element of Y by A1, A2;
A4: ( EqClass x1,PA = F or EqClass x1,PA misses F ) by A1, EQREL_1:def 6;
reconsider x2 = x2 as Element of Y by A1, A3;
x2 in EqClass x2,PA by EQREL_1:def 8;
then EqClass x2,PA meets F by A3, XBOOLE_0:3;
then A5: ( x1 in EqClass x1,PA & EqClass x2,PA = F ) by A1, EQREL_1:def 6, EQREL_1:def 8;

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

suppose A6: for x being Element of Y st x in EqClass x1,PA holds
a . x = TRUE ; :: thesis: (B_INF a,PA) . x1 = (B_INF a,PA) . x2

then (B_INF a,PA) . x1 = TRUE by Def19;
hence (B_INF a,PA) . x1 = (B_INF a,PA) . x2 by A2, A4, A5, A6, Def19, XBOOLE_0:3; :: thesis: verum

end;

suppose A7: ex x being Element of Y st
( x in EqClass x1,PA & not a . x = TRUE ) ; :: thesis: (B_INF a,PA) . x1 = (B_INF a,PA) . x2

then (B_INF a,PA) . x1 = FALSE by Def19;
hence (B_INF a,PA) . x1 = (B_INF a,PA) . x2 by A2, A4, A5, A7, Def19, XBOOLE_0:3; :: thesis: verum

end;

end;

end;

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