reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;

theorem
  for a being Function of Y,BOOLEAN holds a 'xor' O_el(Y) = a
proof
  let a be Function of Y,BOOLEAN;
    let x be Element of Y;
    (a 'xor' O_el(Y)).x =(('not' a '&' O_el(Y)) 'or' (a '&' 'not' O_el(Y))
    ).x by BVFUNC_4:9
      .=(('not' a '&' O_el(Y)) 'or' (a '&' I_el(Y))).x by BVFUNC_1:2
      .=(('not' a '&' O_el(Y)) 'or' a).x by BVFUNC_1:6
      .=(O_el(Y) 'or' a).x by BVFUNC_1:5
      .=a.x by BVFUNC_1:9;
    hence thesis;
end;
