let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds a '&' (O_el Y) = O_el Y
let a be Function of Y,BOOLEAN; :: thesis: a '&' (O_el Y) = O_el Y
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (a '&' (O_el Y)) . x = (O_el Y) . x
thus (a '&' (O_el Y)) . x = (a . x) '&' ((O_el Y) . x) by MARGREL1:def 20
.= (a . x) '&' FALSE by Def10
.= (O_el Y) . x by Def10 ; :: thesis: verum