let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs Y,BOOLEAN holds (a '&' b) '&' c = a '&' (b '&' c)
let a, b, c be Element of Funcs Y,BOOLEAN ; :: thesis: (a '&' b) '&' c = a '&' (b '&' c)
reconsider a9 = a, b9 = b, c9 = c as Element of Funcs Y,BOOLEAN ;
consider k3 being Function such that
A1: (a '&' 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 '&' (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 ((a9 '&' b9) '&' c9) . x = (a9 '&' (b9 '&' c9)) . x

proof

let x be Element of Y; :: thesis: ((a9 '&' b9) '&' c9) . x = (a9 '&' (b9 '&' c9)) . x
A5: ( (a9 . x) '&' ((b . x) '&' (c . x)) = ((a . x) '&' (b . x)) '&' (c . x) & (a9 . x) '&' (b9 . x) = (a9 '&' b9) . x ) by MARGREL1:def 21;
( (a9 '&' (b9 '&' c9)) . x = (a9 . x) '&' ((b9 '&' c9) . x) & (a9 . x) '&' ((b9 '&' c9) . x) = (a9 . x) '&' ((b9 . x) '&' (c9 . x)) ) by MARGREL1:def 21;
hence ((a9 '&' b9) '&' c9) . x = (a9 '&' (b9 '&' c9)) . x by A5, MARGREL1:def 21; :: thesis: verum

end;

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