reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds a 'imp' a '&' a=I_el(Y)
proof
  let a be Function of Y,BOOLEAN;
  for x being Element of Y holds (a 'imp' (a '&' a)).x=TRUE
  proof
    let x be Element of Y;
    (a 'imp' a '&' a).x =TRUE '&' ('not' a.x 'or' a.x) by BVFUNC_1:def 8
      .=TRUE by XBOOLEAN:102;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
