let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds B_INF (a,(%I Y)) = a
let a be Function of Y,BOOLEAN; :: thesis: B_INF (a,(%I Y)) = a
let y be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (B_INF (a,(%I Y))) . y = a . y

A5: now :: thesis: ( ex x being Element of Y st
( x in EqClass (y,(%I Y)) & not a . x = TRUE ) implies not a . y = TRUE )

EqClass (y,(%I Y)) in %I Y ;
then EqClass (y,(%I Y)) in { B where B is Subset of Y : ex z being set st
( B = {z} & z in Y ) } by PARTIT1:31;
then ex B being Subset of Y st
( EqClass (y,(%I Y)) = B & ex z being set st
( B = {z} & z in Y ) ) ;
then consider z being set such that
A6: EqClass (y,(%I Y)) = {z} and
z in Y ;
A7: y in {z} by A6, EQREL_1:def 6;
assume that
A8: ex x being Element of Y st
( x in EqClass (y,(%I Y)) & not a . x = TRUE ) and
A9: a . y = TRUE ; :: thesis: contradiction
consider x1 being Element of Y such that
A10: x1 in EqClass (y,(%I Y)) and
A11: a . x1 <> TRUE by A8;
x1 = z by A10, A6, TARSKI:def 1;
hence contradiction by A9, A11, A7, TARSKI:def 1; :: thesis: verum

end;

A13: now :: thesis: ( ( for x being Element of Y st x in EqClass (y,(%I Y)) holds
a . x = TRUE ) implies (B_INF (a,(%I Y))) . y = a . y )

assume A14: for x being Element of Y st x in EqClass (y,(%I Y)) holds
a . x = TRUE ; :: thesis: (B_INF (a,(%I Y))) . y = a . y
then a . y = TRUE by EQREL_1:def 6;
hence (B_INF (a,(%I Y))) . y = a . y by A14, Def16; :: thesis: verum

end;

now :: thesis: ( ex x being Element of Y st
( x in EqClass (y,(%I Y)) & not a . x = TRUE ) & a . y <> TRUE implies (B_INF (a,(%I Y))) . y = a . y )

assume that
A15: ex x being Element of Y st
( x in EqClass (y,(%I Y)) & not a . x = TRUE ) and
A16: a . y <> TRUE ; :: thesis: (B_INF (a,(%I Y))) . y = a . y
a . y = FALSE by A16, XBOOLEAN:def 3;
hence (B_INF (a,(%I Y))) . y = a . y by A15, Def16; :: thesis: verum

end;

hence (B_INF (a,(%I Y))) . y = a . y by A13, A5; :: thesis: verum