let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b))
let a, b be Function of Y,BOOLEAN; :: thesis: a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b))
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a 'xor' b) . x = ((('not' a) '&' b) 'or' (a '&' ('not' b))) . x
thus (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)))
.= (('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 ; :: thesis: verum