let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ((a '&' b) 'or' c) . x = ((a 'or' c) '&' (b 'or' c)) . x
thus ((a '&' b) 'or' c) . x = ((a '&' b) . x) 'or' (c . x) by Def4
.= ((a . x) '&' (b . x)) 'or' (c . x) by MARGREL1:def 20
.= ((a . x) 'or' (c . x)) '&' ((b . x) 'or' (c . x)) by XBOOLEAN:9
.= ((a . x) 'or' (c . x)) '&' ((b 'or' c) . x) by Def4
.= ((a 'or' c) . x) '&' ((b 'or' c) . x) by Def4
.= ((a 'or' c) '&' (b 'or' c)) . x by MARGREL1:def 20 ; :: thesis: verum