let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds a 'xor' b = ('not' a) 'xor' ('not' b)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: a 'xor' b = ('not' a) 'xor' ('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) 'xor' ('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) 'xor' ('not' b)) . x

proof

let x be Element of Y; :: thesis: (a 'xor' b) . x = (('not' a) 'xor' ('not' b)) . x
(('not' a) 'xor' ('not' b)) . x = ((('not' ('not' a)) '&' ('not' b)) 'or' (('not' a) '&' ('not' ('not' b)))) . x by BVFUNC_4:9
.= (a 'xor' b) . x by BVFUNC_4:9 ;
hence (a 'xor' b) . x = (('not' a) 'xor' ('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) 'xor' ('not' b) by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum