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

theorem Th10:
 for x1,x2 being object holds
  x1<>x2 implies card Choose(X,0,x1,x2)=1
proof let x1,x2 be object;
  assume
A1: x1<>x2;
  per cases;
  suppose
A2: X is empty;
    dom {}=X by A2;
    then reconsider Empty={} as Function of X,{x1,x2} by XBOOLE_1:2;
A3: Choose(X,0,x1,x2) c= {Empty}
    proof
      let z be object;
      assume z in Choose(X,0,x1,x2);
      then consider f be Function of X,{x1,x2} such that
A4:   z=f and
      card (f"{x1})=0 by Def1;
      dom f=X by FUNCT_2:def 1;
      then f=Empty;
      hence thesis by A4,TARSKI:def 1;
    end;
    Empty"{x1}={} & card {}=0;
    then Empty in Choose(X,0,x1,x2) by Def1;
    then Choose(X,0,x1,x2)={Empty} by A3,ZFMISC_1:33;
    hence thesis by CARD_1:30;
  end;
  suppose
A5: X is non empty;
    then consider f be Function such that
A6: dom f=X and
A7: rng f={x2} by FUNCT_1:5;
    rng f c= {x1,x2} by A7,ZFMISC_1:36;
    then
A8: f is Function of X,{x1,x2} by A6,FUNCT_2:2;
A9: Choose(X,0,x1,x2) c= {f}
    proof
      let z be object;
      assume z in Choose(X,0,x1,x2);
      then consider g be Function of X,{x1,x2} such that
A10:  z=g and
A11:  card (g"{x1})=0 by Def1;
      g"{x1}={} by A11;
      then not x1 in rng g by FUNCT_1:72;
      then ( not rng g={x1})& not rng g={x1,x2} by TARSKI:def 1,def 2;
      then dom g=X & rng g={x2} by A5,FUNCT_2:def 1,ZFMISC_1:36;
      then g=f by A6,A7,FUNCT_1:7;
      hence thesis by A10,TARSKI:def 1;
    end;
    not x1 in rng f by A1,A7,TARSKI:def 1;
    then
A12: f"{x1}={} by FUNCT_1:72;
    card {}=0;
    then f in Choose(X,0,x1,x2) by A12,A8,Def1;
    then Choose(X,0,x1,x2)={f} by A9,ZFMISC_1:33;
    hence thesis by CARD_1:30;
  end;
end;
