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

proof

let x be Element of Y; :: thesis: (a 'xor' b) . x = ((a 'or' b) '&' (('not' a) 'or' ('not' b))) . x
((a 'or' b) '&' (('not' a) 'or' ('not' b))) . x = ((a 'or' b) . x) '&' ((('not' a) 'or' ('not' b)) . x) by MARGREL1:def 20
.= ((a . x) 'or' (b . x)) '&' ((('not' a) 'or' ('not' b)) . x) by BVFUNC_1:def 4
.= ((a . x) 'or' (b . x)) '&' ((('not' a) . x) 'or' (('not' b) . x)) by BVFUNC_1:def 4
.= ((('not' a) . x) '&' ((a . x) 'or' (b . x))) 'or' (((a . x) 'or' (b . x)) '&' (('not' b) . x)) by XBOOLEAN:8
.= (((('not' a) . x) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' ((('not' b) . x) '&' ((a . x) 'or' (b . x))) by XBOOLEAN:8
.= (((('not' a) . x) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' ((('not' b) . x) '&' (b . x))) by XBOOLEAN:8
.= ((('not' (a . x)) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' ((('not' b) . x) '&' (b . x))) by MARGREL1:def 19
.= ((('not' (a . x)) '&' (a . x)) 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' (('not' (b . x)) '&' (b . x))) by MARGREL1:def 19
.= (FALSE 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' (('not' (b . x)) '&' (b . x))) by XBOOLEAN:138
.= (FALSE 'or' ((('not' a) . x) '&' (b . x))) 'or' (((('not' b) . x) '&' (a . x)) 'or' FALSE) by XBOOLEAN:138
.= ((('not' a) . x) '&' (b . x)) 'or' (((('not' b) . x) '&' (a . x)) 'or' FALSE) by BINARITH:3
.= ((('not' a) . x) '&' (b . x)) 'or' ((a . x) '&' (('not' b) . x)) by BINARITH:3
.= ((('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
.= (a 'xor' b) . x by BVFUNC_4:9 ;
hence (a 'xor' b) . x = ((a 'or' b) '&' (('not' a) 'or' ('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 = (a 'or' b) '&' (('not' a) 'or' ('not' b)) by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum