let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds a 'or' (O_el Y) = a
let a be Element of Funcs Y,BOOLEAN ; :: thesis: a 'or' (O_el Y) = a
A1: for x being Element of Y holds (a 'or' (O_el Y)) . x = a . x

proof

let x be Element of Y; :: thesis: (a 'or' (O_el Y)) . x = a . x
(a . x) 'or' ((O_el Y) . x) = (a . x) 'or' FALSE by Def13;
hence (a 'or' (O_el Y)) . x = a . x by Def7; :: thesis: verum

end;

consider k3 being Function such that
A2: ( a 'or' (O_el Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( 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, A2, A3;
hence a 'or' (O_el Y) = a by A2, A3, FUNCT_1:9; :: thesis: verum