let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds B_SUP (a,(%O Y)) = B_SUP a
let a be Element of Funcs (Y,BOOLEAN); :: thesis: B_SUP (a,(%O Y)) = B_SUP a
consider k3 being Function such that
A1: B_SUP (a,(%O Y)) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: B_SUP a = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for y being Element of Y holds (B_SUP (a,(%O Y))) . y = (B_SUP a) . y

proof

let y be Element of Y; :: thesis: (B_SUP (a,(%O Y))) . y = (B_SUP a) . y
EqClass (y,(%O Y)) in %O Y ;
then EqClass (y,(%O Y)) in {Y} by PARTIT1:def 9;
then A5: EqClass (y,(%O Y)) = Y by TARSKI:def 1;

A6: now

assume that
A7: for x being Element of Y holds
( not x in EqClass (y,(%O Y)) or not a . x = TRUE ) and
A8: not for x being Element of Y holds a . x = FALSE ; :: thesis: contradiction
consider x1 being Element of Y such that
A9: a . x1 <> FALSE by A8;
a . x1 = TRUE by A9, XBOOLEAN:def 3;
hence contradiction by A5, A7; :: thesis: verum

end;

A10: now

assume that
A11: for x being Element of Y holds
( not x in EqClass (y,(%O Y)) or not a . x = TRUE ) and
A12: for x being Element of Y holds a . x = FALSE ; :: thesis: (B_SUP (a,(%O Y))) . y = (B_SUP a) . y
B_SUP a = O_el Y by A12, Def17;
then (B_SUP a) . y = FALSE by Def13;
hence (B_SUP (a,(%O Y))) . y = (B_SUP a) . y by A11, Def20; :: thesis: verum

end;

now

assume that
A13: ex x being Element of Y st
( x in EqClass (y,(%O Y)) & a . x = TRUE ) and
not for x being Element of Y holds a . x = FALSE ; :: thesis: (B_SUP (a,(%O Y))) . y = (B_SUP a) . y
B_SUP a = I_el Y by A13, Def17;
then (B_SUP a) . y = TRUE by Def14;
hence (B_SUP (a,(%O Y))) . y = (B_SUP a) . y by A13, Def20; :: thesis: verum

end;

hence (B_SUP (a,(%O Y))) . y = (B_SUP a) . y by A10, A6; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence B_SUP (a,(%O Y)) = B_SUP a by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum