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