let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a '&' b) '&' c = a '&' (b '&' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a '&' b) '&' c = a '&' (b '&' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a '&' b) '&' c) . x = (a '&' (b '&' c)) . x
thus ((a '&' b) '&' c) . x = ((a '&' b) . x) '&' (c . x) by MARGREL1:def 20
.= ((a . x) '&' (b . x)) '&' (c . x) by MARGREL1:def 20
.= (a . x) '&' ((b . x) '&' (c . x))
.= (a . x) '&' ((b '&' c) . x) by MARGREL1:def 20
.= (a '&' (b '&' c)) . x by MARGREL1:def 20 ; :: thesis: verum