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

proof

let x be Element of Y; :: 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 21
.= (a . x) '&' ((('not' a) . x) 'or' (b . x)) by BVFUNC_1:def 7
.= ((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 20
.= FALSE 'or' ((a . x) '&' (b . x)) by XBOOLEAN:138
.= (a . x) '&' (b . x) by BINARITH:7
.= (a '&' b) . x by MARGREL1:def 21 ;
hence (a '&' b) . x = (a '&' (('not' a) 'or' b)) . x ; :: thesis: verum

end;

consider k3 being Function such that
A2: ( a '&' b = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( a '&' (('not' a) 'or' b) = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) ) by A1, A2, A3;
hence a '&' b = a '&' (('not' a) 'or' b) by A2, A3, FUNCT_1:9; :: thesis: verum