let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds a '&' ('not' a) = O_el Y

let a be Function of Y,BOOLEAN; :: thesis: a '&' ('not' a) = O_el Y

let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

thus (a '&' ('not' a)) . x = (a . x) '&' (('not' a) . x) by MARGREL1:def 20

.= (a . x) '&' ('not' (a . x)) by MARGREL1:def 19

.= FALSE by XBOOLEAN:138

.= (O_el Y) . x by BVFUNC_1:def 10 ; :: thesis: verum

let a be Function of Y,BOOLEAN; :: thesis: a '&' ('not' a) = O_el Y

let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a '&' ('not' a)) . x = (O_el Y) . x

thus (a '&' ('not' a)) . x = (a . x) '&' (('not' a) . x) by MARGREL1:def 20

.= (a . x) '&' ('not' (a . x)) by MARGREL1:def 19

.= FALSE by XBOOLEAN:138

.= (O_el Y) . x by BVFUNC_1:def 10 ; :: thesis: verum