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

proof

let x be Element of Y; :: thesis: (a 'xor' (O_el Y)) . x = a . x
(a 'xor' (O_el Y)) . x = ((('not' a) '&' (O_el Y)) 'or' (a '&' ('not' (O_el Y)))) . x by BVFUNC_4:9
.= ((('not' a) '&' (O_el Y)) 'or' (a '&' (I_el Y))) . x by BVFUNC_1:2
.= ((('not' a) '&' (O_el Y)) 'or' a) . x by BVFUNC_1:6
.= ((O_el Y) 'or' a) . x by BVFUNC_1:5
.= a . x by BVFUNC_1:9 ;
hence (a 'xor' (O_el Y)) . x = a . x ; :: thesis: verum

end;

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