let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
A1: for x being Element of Y holds ((a '&' b) 'or' c) . x = ((a 'or' c) '&' (b 'or' c)) . x

proof

let x be Element of Y; :: thesis: ((a '&' b) 'or' c) . x = ((a 'or' c) '&' (b 'or' c)) . x
((a '&' b) 'or' c) . x = ((a '&' b) . x) 'or' (c . x) by Def7;
then A2: ((a '&' b) 'or' c) . x = ((a . x) '&' (b . x)) 'or' (c . x) by MARGREL1:def 21;
A3: (c . x) 'or' ((a . x) '&' (b . x)) = ((c . x) 'or' (a . x)) '&' ((c . x) 'or' (b . x)) by XBOOLEAN:9;
(a . x) 'or' (c . x) = (a 'or' c) . x by Def7;
then ((a '&' b) 'or' c) . x = ((a 'or' c) . x) '&' ((b 'or' c) . x) by A2, A3, Def7
.= ((a 'or' c) '&' (b 'or' c)) . x by MARGREL1:def 21 ;
hence ((a '&' b) 'or' c) . x = ((a 'or' c) '&' (b 'or' c)) . x ; :: thesis: verum

end;

consider k3 being Function such that
A4: ( (a '&' b) 'or' c = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A5: ( (a 'or' c) '&' (b 'or' c) = 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, A4, A5;
hence (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c) by A4, A5, FUNCT_1:9; :: thesis: verum