let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
consider k3 being Function such that
A1: (a 'or' b) '&' c = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a '&' c) 'or' (b '&' c) = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((a 'or' b) '&' c) . x = ((a '&' c) 'or' (b '&' c)) . x

proof

let x be Element of Y; :: thesis: ((a 'or' b) '&' c) . x = ((a '&' c) 'or' (b '&' c)) . x
((a 'or' b) '&' c) . x = ((a 'or' b) . x) '&' (c . x) by MARGREL1:def 21;
then A5: ((a 'or' b) '&' c) . x = ((a . x) 'or' (b . x)) '&' (c . x) by Def7;
( (c . x) '&' ((a . x) 'or' (b . x)) = ((c . x) '&' (a . x)) 'or' ((c . x) '&' (b . x)) & (a . x) '&' (c . x) = (a '&' c) . x ) by MARGREL1:def 21, XBOOLEAN:8;
then ((a 'or' b) '&' c) . x = ((a '&' c) . x) 'or' ((b '&' c) . x) by A5, MARGREL1:def 21
.= ((a '&' c) 'or' (b '&' c)) . x by Def7 ;
hence ((a 'or' b) '&' c) . x = ((a '&' c) 'or' (b '&' c)) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c) by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum