reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th27:
 for x being object  holds
  X,{x} are_equipotent iff ex x being object st X = { x }
proof let x be object;
  thus X,{x} are_equipotent implies ex x being object st X = { x }
  proof
    assume X,{x} are_equipotent;
    then consider f such that
    f is one-to-one and
A1: dom f = { x } and
A2: rng f = X by WELLORD2:def 4;
    rng f = { f.x } by A1,FUNCT_1:4;
    hence thesis by A2;
  end;
  given y being object such that
A3: X = { y };
  take f = X --> x;
A4: dom f = X;
  thus f is one-to-one
  proof
    let a,b be object;
    assume that
A5: a in dom f and
A6: b in dom f and
    f.a = f.b;
    a = y by A3,A5,TARSKI:def 1;
    hence thesis by A3,A6,TARSKI:def 1;
  end;
  thus dom f = X;
  thus rng f c= { x } by FUNCOP_1:13;
  let a be object;
  assume a in { x };
  then
A7: a = x by TARSKI:def 1;
A8: y in { y } by TARSKI:def 1;
  then f.y = x by A3,FUNCOP_1:7;
  hence thesis by A3,A4,A7,A8,FUNCT_1:def 3;
end;
