let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds B_INF a,(%O Y) = B_INF a
let a be Element of Funcs Y,BOOLEAN ; :: thesis: B_INF a,(%O Y) = B_INF a
A1: for y being Element of Y holds (B_INF a,(%O Y)) . y = (B_INF a) . y

proof

let y be Element of Y; :: thesis: (B_INF a,(%O Y)) . y = (B_INF a) . y

A2: now

assume A3: ( ( for x being Element of Y holds a . x = TRUE ) & ( for x being Element of Y st x in EqClass y,(%O Y) holds
a . x = TRUE ) ) ; :: thesis: (B_INF a,(%O Y)) . y = (B_INF a) . y
then B_INF a = I_el Y by Def16;
then (B_INF a) . y = TRUE by Def14;
hence (B_INF a,(%O Y)) . y = (B_INF a) . y by A3, Def19; :: thesis: verum

end;

A4: now

assume A5: ( not for x being Element of Y holds a . x = TRUE & ( for x being Element of Y st x in EqClass y,(%O Y) holds
a . x = TRUE ) ) ; :: thesis: contradiction
EqClass y,(%O Y) in %O Y ;
then EqClass y,(%O Y) in {Y} by PARTIT1:def 9;
then EqClass y,(%O Y) = Y by TARSKI:def 1;
hence contradiction by A5; :: thesis: verum

end;

now

assume A6: ( not for x being Element of Y holds a . x = TRUE & ex x being Element of Y st
( x in EqClass y,(%O Y) & not a . x = TRUE ) ) ; :: thesis: (B_INF a,(%O Y)) . y = (B_INF a) . y
then B_INF a = O_el Y by Def16;
then (B_INF a) . y = FALSE by Def13;
hence (B_INF a,(%O Y)) . y = (B_INF a) . y by A6, Def19; :: thesis: verum

end;

hence (B_INF a,(%O Y)) . y = (B_INF a) . y by A2, A4; :: thesis: verum

end;

consider k3 being Function such that
A7: ( B_INF a,(%O Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A8: ( B_INF a = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A7, A8;
hence B_INF a,(%O Y) = B_INF a by A7, A8, FUNCT_1:9; :: thesis: verum