let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a 'or' b) '&' c) . x = ((a '&' c) 'or' (b '&' c)) . x
thus ((a 'or' b) '&' c) . x = ((a 'or' b) . x) '&' (c . x) by MARGREL1:def 20
.= ((a . x) 'or' (b . x)) '&' (c . x) by Def4
.= ((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x)) by XBOOLEAN:8
.= ((a . x) '&' (c . x)) 'or' ((b '&' c) . x) by MARGREL1:def 20
.= ((a '&' c) . x) 'or' ((b '&' c) . x) by MARGREL1:def 20
.= ((a '&' c) 'or' (b '&' c)) . x by Def4 ; :: thesis: verum