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
A1: 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
A2: (a . x) '&' FALSE = FALSE ;
FALSE = (O_el Y) . x by Def13;
hence (a '&' (O_el Y)) . x = (O_el Y) . x by A2, MARGREL1:def 21; :: thesis: verum

end;

consider k3 being Function such that
A3: ( a '&' (O_el Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A4: ( O_el Y = 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, A3, A4;
hence a '&' (O_el Y) = O_el Y by A3, A4, FUNCT_1:9; :: thesis: verum