reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  dom f = X implies product f,product (f*P) are_equipotent
proof
  assume
A1: dom f = X;
A2: rng P = X by FUNCT_2:def 3;
  dom P = X by FUNCT_2:52;
  then
A3: dom (f*P) = X by A1,A2,RELAT_1:27;
A4: rng (P") = X by FUNCT_2:def 3;
A5: dom (P") = X by FUNCT_2:52;
  now
    let x;
    assume
A6: x in dom (P");
    then P".x in X by A4,FUNCT_1:def 3;
    then (f*P).(P".x) = f.(P.(P".x)) by A3,FUNCT_1:12
      .= f.x by A5,A2,A6,FUNCT_1:35;
    hence f.x,(f*P).(P".x) are_equipotent;
  end;
  hence thesis by A1,A5,A4,A3,Th39;
end;
