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

theorem Th2:
  for X,Y,x,y st not x in X & y in Y holds card Funcs(X,Y) = card{F
  where F is Function of X\/{x},Y:F.x=y}
proof
  let X,Y,x,y such that
A1: not x in X and
A2: y in Y;
  set F2={F where F is Function of X\/{x},Y:F.x=y};
  set F1={F where F is Function of X\/{x},Y\/{y}:rng (F|X) c=Y & F.x=y};
  {y} c= Y by A2,ZFMISC_1:31;
  then
A3: Y=Y\/{y} by XBOOLE_1:12;
A4: F2 c= F1
  proof
    let f be object;
    assume f in F2;
    then consider F be Function of X\/{x},Y such that
A5: f=F & F.x=y;
    rng (F|X) c=Y;
    hence thesis by A3,A5;
  end;
  F1 c= F2
  proof
    let f be object;
    assume f in F1;
    then ex F be Function of X\/{x},Y\/{y} st f=F & rng (F|X) c=Y & F.x=y;
    hence thesis by A3;
  end;
  then F1=F2 by A4;
  hence thesis by A1,A2,Th1;
end;
