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

proof

let x be Element of Y; :: thesis: (a '&' (O_el Y)) . x = (O_el Y) . x
( (a . x) '&' FALSE = FALSE & FALSE = (O_el Y) . x ) by Def13;
hence (a '&' (O_el Y)) . x = (O_el Y) . x by MARGREL1:def 20; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a '&' (O_el Y) = O_el Y by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum