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

theorem Th11:
 for x1,x2 being object holds
  card Choose(X,card X,x1,x2)=1
proof let x1,x2 be object;
  per cases;
  suppose
A1: X is empty;
    dom {}=X by A1;
    then reconsider Empty={} as Function of X,{x1,x2} by XBOOLE_1:2;
A2: Choose(X,card X,x1,x2) c= {Empty}
    proof
      let z be object;
      assume z in Choose(X,card X,x1,x2);
      then consider f be Function of X,{x1,x2} such that
A3:   z=f and
      card (f"{x1})=card X by Def1;
      dom f=X by FUNCT_2:def 1;
      then f=Empty;
      hence thesis by A3,TARSKI:def 1;
    end;
    Empty"{x1}={};
    then Empty in Choose(X,card X,x1,x2) by A1,Def1;
    then Choose(X,card X,x1,x2)={Empty} by A2,ZFMISC_1:33;
    hence thesis by CARD_1:30;
  end;
  suppose
A4: X is non empty;
    then consider f be Function such that
A5: dom f=X and
A6: rng f={x1} by FUNCT_1:5;
    rng f c= {x1,x2} by A6,ZFMISC_1:36;
    then
A7: f is Function of X,{x1,x2} by A5,FUNCT_2:2;
A8: Choose(X,card X,x1,x2) c= {f}
    proof
      let z be object;
      assume z in Choose(X,card X,x1,x2);
      then consider g be Function of X,{x1,x2} such that
A9:  z=g and
A10:  card (g"{x1})=card X by Def1;
A11:  now
        per cases;
        suppose
          x1=x2;
          then {x1,x2}={x1} by ENUMSET1:29;
          hence rng g={x1} by A4,ZFMISC_1:33;
        end;
        suppose
A12:      x1<>x2;
          g"{x2}={}
          proof
            assume g"{x2}<>{};
            then consider z being object such that
A13:        z in g"{x2} by XBOOLE_0:def 1;
            g.z in {x2} by A13,FUNCT_1:def 7;
            then
A14:        g.z=x2 by TARSKI:def 1;
            g"{x1}=X by A10,CARD_2:102;
            then g.z in {x1} by A13,FUNCT_1:def 7;
            hence thesis by A12,A14,TARSKI:def 1;
          end;
          then not x2 in rng g by FUNCT_1:72;
          then ( not rng g={x2})& not rng g={x1,x2} by TARSKI:def 1,def 2;
          hence rng g={x1} by A4,ZFMISC_1:36;
        end;
      end;
      dom g=X by FUNCT_2:def 1;
      then g=f by A5,A6,A11,FUNCT_1:7;
      hence thesis by A9,TARSKI:def 1;
    end;
    card (f"{x1})=card X by A5,A6,RELAT_1:134;
    then f in Choose(X,card X,x1,x2) by A7,Def1;
    then Choose(X,card X,x1,x2)={f} by A8,ZFMISC_1:33;
    hence thesis by CARD_1:30;
  end;
end;
