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

theorem Th7:
  card{F where F is Function of X,X:F is Permutation of X}=card X!
proof
  set F1={F where F is Function of X,X:F is one-to-one};
  set F2={F where F is Function of X,X:F is Permutation of X};
A1: F1 c= F2
  proof
    let x be object;
    assume x in F1;
    then consider F be Function of X,X such that
A2: x=F and
A3: F is one-to-one;
    card X=card X;
    then F is onto by A3,FINSEQ_4:63;
    hence thesis by A2,A3;
  end;
  (card X-'card X)!=1 by NEWTON:12,XREAL_1:232;
  then
A4: card X!/((card X-'card X)!)=card X!;
  F2 c= F1
  proof
    let x be object;
    assume x in F2;
    then ex F be Function of X,X st x=F & F is Permutation of X;
    hence thesis;
  end;
  then F1=F2 by A1;
  hence thesis by A4,Th6;
end;
