
theorem Th17:
  for X being set ex f being Function of BoolePoset X, (BoolePoset
  {0})|^X st f is isomorphic & for Y being Subset of X holds f.Y = chi(Y,X)
proof
  let Z be set;
  per cases;
  suppose
A1: Z = {};
    then
A2: (BoolePoset{0})|^Z = RelStr(#{{}}, id {{}}#) by YELLOW_1:27;
A3: InclPoset bool {} = RelStr(#bool {}, RelIncl bool {}#) by YELLOW_1:def 1;
A4: BoolePoset Z = InclPoset bool {} by A1,YELLOW_1:4;
    then reconsider
    f = id {{}} as Function of BoolePoset Z, (BoolePoset{0})|^Z by A3,A1,
YELLOW_1:27,ZFMISC_1:1;
    take f;
A5: rng id {{}} = {{}};
    for x,y being Element of BoolePoset Z holds x <= y iff f.x <= f.y by A4,A3;
    hence f is isomorphic by A2,A5,WAYBEL_0:66;
    let Y be Subset of Z;
    Y = {} by A1;
    then Y in {{}} by TARSKI:def 1;
    then f.Y = {} by A1,FUNCT_1:18;
    hence thesis by A1;
  end;
  suppose
    Z <> {};
    then reconsider Z9 = Z as non empty set;
    (BoolePoset{0})|^Z = product (Z9 --> BoolePoset{0}) by YELLOW_1:def 5;
    hence thesis by WAYBEL18:14;
  end;
end;
