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
  for x,y be set,X be finite set, P be Function of card Choose(X,k,x,y),
Choose(X,k,x,y) st dom Fy=X & P is one-to-one & x<>y for XFS be XFinSequence of
  NAT st dom XFS=dom P & (for z,f st z in dom XFS & f=P.z holds XFS.z=card(
  Intersection(Fy,f,x))) holds Card_Intersection(Fy,k)=Sum XFS
proof
  let x,y be set,X be finite set, P be Function of card Choose(X,k,x,y),Choose
  (X,k,x,y);
  assume dom Fy=X & P is one-to-one & x<>y;
  then consider XFS be XFinSequence of NAT such that
A1: dom XFS=dom P and
A2: for z,f st z in dom XFS & f=P.z holds XFS.z=card(Intersection(Fy,f,x )) and
A3: Card_Intersection(Fy,k)=Sum XFS by Def3;
  let XFS1 be XFinSequence of NAT such that
A4: dom XFS1=dom P and
A5: for z,f st z in dom XFS1 & f=P.z holds XFS1.z=card(Intersection(Fy,f ,x));
  now
    let z be object such that
A6: z in dom XFS;
    P.z in rng P by A1,A6,FUNCT_1:def 3;
    then consider Pz be Function of X,{x,y} such that
A7: Pz=P.z and
    card (Pz"{x})=k by Def1;
    XFS1.z=card(Intersection(Fy,Pz,x)) by A4,A5,A1,A6,A7;
    hence XFS1.z=XFS.z by A2,A6,A7;
  end;
  hence thesis by A4,A1,A3,FUNCT_1:2;
end;
