
theorem Th9:
  for f being Function, x being set st x in dom f
  holds f orbit (f.x) c= f orbit x
proof
  let f be Function;
  let x be set;
  assume that
A1: x in dom f;
  let a be object;
  assume a in f orbit (f.x);
  then consider n being Element of NAT such that
A2: a = iter(f,n).(f.x) and
A3: f.x in dom iter(f,n);
A4: iter(f,n+1) = iter(f,n)*f by FUNCT_7:69;
  then
A5: a = iter(f,n+1).x by A1,A2,FUNCT_1:13;
  x in dom iter(f,n+1) by A1,A3,A4,FUNCT_1:11;
  hence thesis by A5;
end;
