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

proof

let x be Element of Y; :: thesis: (a 'or' b) . x = (a 'or' (('not' a) '&' b)) . x
(a 'or' (('not' a) '&' b)) . x = (a . x) 'or' ((('not' a) '&' b) . x) by BVFUNC_1:def 7
.= (a . x) 'or' ((('not' a) . x) '&' (b . x)) by MARGREL1:def 21
.= ((a . x) 'or' (('not' a) . x)) '&' ((a . x) 'or' (b . x)) by XBOOLEAN:9
.= ((a . x) 'or' ('not' (a . x))) '&' ((a . x) 'or' (b . x)) by MARGREL1:def 20
.= TRUE '&' ((a . x) 'or' (b . x)) by XBOOLEAN:102
.= (a . x) 'or' (b . x) by MARGREL1:50
.= (a 'or' b) . x by BVFUNC_1:def 7 ;
hence (a 'or' b) . x = (a 'or' (('not' a) '&' b)) . x ; :: thesis: verum

end;

consider k3 being Function such that
A2: ( a 'or' b = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( a 'or' (('not' a) '&' 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 'or' b = a 'or' (('not' a) '&' b) by A2, A3, FUNCT_1:9; :: thesis: verum