let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds 'not' (a 'or' b) = ('not' a) '&' ('not' b)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: 'not' (a 'or' b) = ('not' a) '&' ('not' b)
consider k3 being Function such that
A1: 'not' (a 'or' b) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: ('not' a) '&' ('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 ('not' (a 'or' b)) . x = (('not' a) '&' ('not' b)) . x

proof

let x be Element of Y; :: thesis: ('not' (a 'or' b)) . x = (('not' a) '&' ('not' b)) . x
(a 'or' b) . x = (a . x) 'or' (b . x) by Def7;
then A5: ('not' (a 'or' b)) . x = 'not' ((a . x) 'or' (b . x)) by MARGREL1:def 20
.= ('not' (a . x)) '&' ('not' (b . x)) ;
'not' (a . x) = ('not' a) . x by MARGREL1:def 20;
then ('not' (a 'or' b)) . x = (('not' a) . x) '&' (('not' b) . x) by A5, MARGREL1:def 20
.= (('not' a) '&' ('not' b)) . x by MARGREL1:def 21 ;
hence ('not' (a 'or' b)) . x = (('not' a) '&' ('not' b)) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence 'not' (a 'or' b) = ('not' a) '&' ('not' b) by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum