
theorem Th10:
  for f,g being Function, a,A being set st rng f c= dom f & a in dom f holds
  not f orbit a c= A implies
  ex n being Nat st ((A,g) iter f).a = iter(f,n).a & iter(f,n).a nin A &
  for i being Nat st i < n holds iter(f,i).a in A
proof
  let f,g be Function;
  let a,A be set;
  assume
A1: rng f c= dom f;
  assume
A2: a in dom f;
  assume not f orbit a c= A;
  then consider y being object such that
A3: y in f orbit a and
A4: y nin A;
A5: ex n1 being Element of NAT st ( y = iter(f,n1).a)&( a in dom
  iter(f,n1)) by A3;
  defpred R[Nat] means iter(f,$1).a nin A;
A6: ex n being Nat st R[n] by A4,A5;
  consider n being Nat such that
A7: R[n] and
A8: for m being Nat st R[m] holds n <= m from NAT_1:sch 5(A6);
  take n;
  for i being Nat holds i < n implies iter(f,i).a in A by A8;
  hence ((A,g) iter f).a = iter(f,n).a by A1,A2,A7,Def7;
  thus thesis by A7,A8;
end;
