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

theorem
  card X<>k implies Choose(X,k,x1,x1) is empty
proof
  assume
A1: card X<>k;
  assume Choose(X,k,x1,x1) is non empty;
  then consider y being object such that
A2: y in Choose(X,k,x1,x1);
  consider f be Function of X,{x1,x1} such that
  f = y and
A3: card (f"{x1}) = k by A2,Def1;
  per cases;
  suppose
A4: rng f is empty;
A5: dom f=X by FUNCT_2:def 1;
    dom f={} by A4,RELAT_1:42;
    hence thesis by A1,A3,A5;
  end;
  suppose
A6: rng f is non empty;
    {x1,x1}={x1} by ENUMSET1:29;
    then rng f={x1} by A6,ZFMISC_1:33;
    then f"{x1}=dom f by RELAT_1:134;
    hence thesis by A1,A3,FUNCT_2:def 1;
  end;
end;
