
theorem
  for f being Function, x,y being set
  st rng f c= dom f & y in f orbit x holds f.y in f orbit x
proof
  let f be Function;
  let x,y be set;
  assume
A1: rng f c= dom f;
  assume y in f orbit x;
  then consider n being Element of NAT such that
A2: y = iter(f,n).x and
A3: x in dom iter(f,n);
A4: iter(f,n+1) = f*iter(f,n) by FUNCT_7:71;
  then
A5: f.y = iter(f,n+1).x by A2,A3,FUNCT_1:13;
A6: y in rng iter(f,n) by A2,A3,FUNCT_1:def 3;
A7: rng iter(f,n) c= field f by FUNCT_7:72;
  field f = dom f by A1,XBOOLE_1:12;
  then x in dom iter(f,n+1) by A2,A3,A4,A6,A7,FUNCT_1:11;
  hence thesis by A5;
end;
