let Y be non empty set ; :: thesis: for a, b being Element of Funcs (Y,BOOLEAN) holds a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b))
let a, b be Element of Funcs (Y,BOOLEAN); :: thesis: a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b))
consider k3 being Function such that
A1: a 'xor' b = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: (('not' a) '&' b) 'or' (a '&' ('not' b)) = 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' b) . x = ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x

proof

let x be Element of Y; :: thesis: (a 'xor' b) . x = ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x
(a 'xor' b) . x = (a . x) 'xor' (b . x) by BVFUNC_1:def 5
.= ((('not' a) . x) '&' (b . x)) 'or' ((a . x) '&' ('not' (b . x))) by MARGREL1:def 19
.= ((('not' a) . x) '&' (b . x)) 'or' ((a . x) '&' (('not' b) . x)) by MARGREL1:def 19
.= ((('not' a) '&' b) . x) 'or' ((a . x) '&' (('not' b) . x)) by MARGREL1:def 20
.= ((('not' a) '&' b) . x) 'or' ((a '&' ('not' b)) . x) by MARGREL1:def 20
.= ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x by BVFUNC_1:def 4 ;
hence (a 'xor' b) . x = ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b)) by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum