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 '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
proof
  let a,b,c be Function of Y,BOOLEAN;
  let x be Element of Y;
  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;
end;
