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

theorem Th16:
 for x,y being object holds
  x<>y implies (Y-->y)+*(X-->x) in Choose(X\/Y,card X,x,y)
proof let x,y be object;
  set F=(Y-->y)+*(X-->x);
  dom (Y-->y)=Y & dom (X-->x)=X;
  then
A1: dom F=X\/Y by FUNCT_4:def 1;
  {y} c= {x,y} by ZFMISC_1:7;
  then
A2: rng (Y-->y) c= {x,y};
  {x} c= {x,y} by ZFMISC_1:7;
  then rng (X-->x)c= {x,y};
  then rng F c= rng(X-->x)\/rng(Y-->y) & rng(X-->x)\/rng(Y-->y)c={x,y} by A2,
FUNCT_4:17,XBOOLE_1:8;
  then reconsider F as Function of X\/Y,{x,y} by A1,FUNCT_2:2,XBOOLE_1:1;
  assume
A3: x<>y;
A4: F"{x}c=X
  proof
    let z be object such that
A5: z in F"{x};
A6: z in X or z in Y by A5,XBOOLE_0:def 3;
    F.z in {x} by A5,FUNCT_1:def 7;
    then
A7: F.z=x by TARSKI:def 1;
    z in dom F by A5,FUNCT_1:def 7;
    then
A8: z in dom (X-->x)\/dom (Y-->y);
    assume
A9: not z in X;
    F.z=(Y-->y).z by A9,A8,FUNCT_4:def 1;
    hence contradiction by A3,A9,A6,A7,FUNCOP_1:7;
  end;
  X c= F"{x}
  proof
    let z be object such that
A10: z in X;
A11: z in dom F by A1,A10,XBOOLE_0:def 3;
    z in dom (X-->x) by A10;
    then
A12: F.z=(X-->x).z by FUNCT_4:13;
    (X-->x).z=x by A10,FUNCOP_1:7;
    then F.z in {x} by A12,TARSKI:def 1;
    hence thesis by A11,FUNCT_1:def 7;
  end;
  then X=F"{x} by A4;
  hence thesis by Def1;
end;
