reserve Y for non empty set;

theorem Th5:
  for a being Function of Y,BOOLEAN holds a '&' 'not' a = O_el(Y)
proof
  let a be Function of Y,BOOLEAN;
  let x be Element of Y;
  thus (a '&' 'not' a).x = a.x '&' ('not' a).x by MARGREL1:def 20
     .= a.x '&' 'not' a.x by MARGREL1:def 19
     .= FALSE by XBOOLEAN:138
     .= O_el(Y).x by BVFUNC_1:def 10;
end;
