reserve Y for non empty set;
reserve B for Subset of Y;

theorem ::Distributive
  for a,b,c being Function of Y,BOOLEAN
  holds (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
proof
  let a,b,c be Function of Y,BOOLEAN;
  let x be Element of Y;
  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;
end;
