reserve f, g, h for Function;
reserve x, y, z, u, X for set,
  A for non empty set,
  n for Element of NAT,
  f for Function of X, X;

theorem Th2:
  x is_a_fixpoint_of iter(f,n) implies f.x is_a_fixpoint_of iter(f, n)
proof
  assume
A1: x is_a_fixpoint_of iter(f,n);
  then
A2: x in dom iter(f, n);
A3: dom f = X by FUNCT_2:52;
  then dom iter(f, n) = X & f.x in rng f by A2,FUNCT_1:def 3,FUNCT_2:52;
  hence f.x in dom iter(f, n);
  x = iter(f, n).x by A1;
  hence f.x = (f*iter(f, n)).x by A2,FUNCT_1:13
    .= iter(f, n+1).x by FUNCT_7:71
    .= (iter(f, n)*f).x by FUNCT_7:69
    .= iter(f, n).(f.x) by A2,A3,FUNCT_1:13;
end;
