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 Th56:
  for Fy,X,n,k st dom Fy=X holds (ex x,y st x<>y & for f st f in
Choose(X,k,x,y) holds card(Intersection(Fy,f,x))=n) implies Card_Intersection(
  Fy,k)=n*(card X choose k)
proof
  let Fy,X,n,k such that
A1: X=dom Fy;
  assume ex x,y st x<>y & for f st f in Choose(X,k,x,y) holds card(
  Intersection(Fy,f,x))=n;
  then consider x,y such that
A2: x<>y and
A3: for f st f in Choose(X,k,x,y) holds card(Intersection(Fy,f,x))=n;
  set Ch=Choose(X,k,x,y);
  consider P be Function of card Ch,Ch such that
A4: P is one-to-one by Lm2;
  consider XFS be XFinSequence of NAT such that
A5: dom XFS=dom P and
A6: for z,f st z in dom XFS & f=P.z holds XFS.z=card(Intersection(Fy,f,x )) and
A7: Card_Intersection(Fy,k)=Sum XFS by A1,A2,A4,Def3;
  for z being object st z in dom XFS holds XFS.z = n
  proof
    let z be object such that
A8: z in dom XFS;
A9: P.z in rng P by A5,A8,FUNCT_1:def 3;
    then consider f be Function of X,{x,y} such that
A10: f=P.z and
    card (f"{x})=k by Def1;
    XFS.z=card(Intersection(Fy,f,x)) by A6,A8,A10;
    hence thesis by A3,A9,A10;
  end;
  then
A11: XFS=dom XFS-->n by FUNCOP_1:11;
  then
A12: rng XFS c= {n} by FUNCOP_1:13;
  Ch={} implies card Ch={};
  then
A13: dom P=card Ch by FUNCT_2:def 1;
  n in {n} by TARSKI:def 1;
  then {n} c= {0,n} & XFS"{n}=dom P by A5,A11,FUNCOP_1:14,ZFMISC_1:7;
  then Sum XFS=n*card card Ch by A12,A13,AFINSQ_2:68,XBOOLE_1:1;
  hence thesis by A2,A7,Th15;
end;
