let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a '&' b = a '&' (('not' a) 'or' b)
let a, b be Function of Y,BOOLEAN; :: thesis: a '&' b = a '&' (('not' a) 'or' b)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a '&' b) . x = (a '&' (('not' a) 'or' b)) . x
(a '&' (('not' a) 'or' b)) . x = (a . x) '&' ((('not' a) 'or' b) . x) by MARGREL1:def 20
.= (a . x) '&' ((('not' a) . x) 'or' (b . x)) by BVFUNC_1:def 4
.= ((a . x) '&' (('not' a) . x)) 'or' ((a . x) '&' (b . x)) by XBOOLEAN:8
.= ((a . x) '&' ('not' (a . x))) 'or' ((a . x) '&' (b . x)) by MARGREL1:def 19
.= FALSE 'or' ((a . x) '&' (b . x)) by XBOOLEAN:138
.= (a . x) '&' (b . x) by BINARITH:3
.= (a '&' b) . x by MARGREL1:def 20 ;
hence (a '&' b) . x = (a '&' (('not' a) 'or' b)) . x ; :: thesis: verum