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 Th54:
  for Fy be finite-yielding Function,X ex XFS be XFinSequence of
  INT st dom XFS = card X & for n st n in dom XFS holds XFS.n=((-1)|^n)*
  Card_Intersection(Fy,n+1)
proof
  let Fy be finite-yielding Function,X;
  defpred P[set,set] means for n st n=$1 holds $2=((-1)|^n)*Card_Intersection(
  Fy,n+1);
A1: for k st k in Segm card X ex x be Element of INT st P[k,x]
  proof
    let k such that
    k in Segm card X;
    reconsider C=((-1)|^k)*Card_Intersection(Fy,k+1) as Element of INT
        by INT_1:def 2;
    take C;
    thus thesis;
  end;
  consider XFS be XFinSequence of INT such that
A2: dom XFS = Segm card X & for k st k in Segm card X holds P[k,XFS.k] from
  STIRL2_1:sch 5(A1);
  take XFS;
  thus thesis by A2;
end;
