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

proof

let x be Element of Y; :: thesis: (a 'or' (I_el Y)) . x = (I_el Y) . x
A2: (a . x) 'or' TRUE = TRUE ;
TRUE = (I_el Y) . x by Def14;
hence (a 'or' (I_el Y)) . x = (I_el Y) . x by A2, Def7; :: thesis: verum

end;

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