let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds 'not' (a '&' b) = ('not' a) 'or' ('not' b)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: 'not' (a '&' b) = ('not' a) 'or' ('not' b)
A1: for x being Element of Y holds ('not' (a '&' b)) . x = (('not' a) 'or' ('not' b)) . x

proof

let x be Element of Y; :: thesis: ('not' (a '&' b)) . x = (('not' a) 'or' ('not' b)) . x
(a '&' b) . x = (a . x) '&' (b . x) by MARGREL1:def 21;
then A2: ('not' (a '&' b)) . x = ('not' (a . x)) 'or' ('not' (b . x)) by MARGREL1:def 20;
'not' (a . x) = ('not' a) . x by MARGREL1:def 20;
then ('not' (a '&' b)) . x = (('not' a) . x) 'or' (('not' b) . x) by A2, MARGREL1:def 20
.= (('not' a) 'or' ('not' b)) . x by Def7 ;
hence ('not' (a '&' b)) . x = (('not' a) 'or' ('not' b)) . x ; :: thesis: verum

end;

consider k3 being Function such that
A3: ( 'not' (a '&' b) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A4: ( ('not' a) 'or' ('not' b) = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A3, A4;
hence 'not' (a '&' b) = ('not' a) 'or' ('not' b) by A3, A4, FUNCT_1:9; :: thesis: verum