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;
reserve k,n,m for Nat;
reserve l for Element of omega;

theorem
 for x being object holds
  X,[:X,{x}:] are_equipotent & card X = card [:X,{x}:]
proof let x be object;
  deffunc f(object) = [$1,x];
  consider f such that
A1: dom f = X & for y being object st y in X holds f.y = f(y)
from FUNCT_1:sch 3;
  thus X,[:X,{x}:] are_equipotent
  proof
    take f;
    thus f is one-to-one
    proof
      let y,z be object;
      assume that
A2:   y in dom f & z in dom f and
A3:   f.y = f.z;
A4:   [y,x]`1 = y & [z,x]`1 = z;
      f.y = [y,x] & f.z = [z,x] by A1,A2;
      hence y = z by A3,A4;
    end;
    thus dom f = X by A1;
    thus rng f c= [:X,{x}:]
    proof
      let y be object;
A5:   x in {x} by TARSKI:def 1;
      assume y in rng f;
      then consider z being object such that
A6:   z in dom f and
A7:   y = f.z by FUNCT_1:def 3;
      y = [z,x] by A1,A6,A7;
      hence thesis by A1,A6,A5,ZFMISC_1:87;
    end;
    let y be object;
    assume y in [:X,{x}:];
    then consider y1,y2 being object such that
A8: y1 in X and
A9: y2 in {x} and
A10: y = [y1,y2] by ZFMISC_1:84;
    y2 = x by A9,TARSKI:def 1;
    then y = f.y1 by A1,A8,A10;
    hence thesis by A1,A8,FUNCT_1:def 3;
  end;
  hence thesis by Th4;
end;
