reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;
reserve Fy for finite-yielding Function;

theorem
  ex XF be XFinSequence of INT st Sum XF=card {h where h is Function of
X,X: h is one-to-one & for x st x in X holds h.x<>x}& dom XF = card X+1 & for n
  st n in dom XF holds XF.n=((-1)|^n)*(card X!)/(n!)
proof
  set S1={h where h is Function of X,X: h is one-to-one & for x st x in X
  holds h.x<>(id X).x};
  set S2={h where h is Function of X,X: h is one-to-one & for x st x in X
  holds h.x<>x};
A1: S2 c= S1
  proof
    let x be object;
    assume x in S2;
    then consider h be Function of X,X such that
A2: h=x & h is one-to-one and
A3: for y st y in X holds h.y<>y;
     for y st y in X holds (id X).y<>h.y by A3,FUNCT_1:17;
    hence thesis by A2;
  end;
A4: dom id X=X & rng id X=X;
  S1 c= S2
  proof
    let x be object;
    assume x in S1;
    then consider h be Function of X,X such that
A5: h=x & h is one-to-one and
A6: for y st y in X holds h.y<>(id X).y;
    now
      let y such that
A7:   y in X;
      (id X).y=y by A7,FUNCT_1:17;
      hence h.y<>y by A6,A7;
    end;
    hence thesis by A5;
  end;
  then S1=S2 by A1;
  hence thesis by A4,Th61;
end;
