Proof that If G = (V , E ) is a graph, then S is an independent set ⇐⇒ V − S is a vertex cover. I have the following :
  Proof:
Proof.  ⇒ Suppose $S$ is an independent set, and let $ e = (u, v )$
be some edge. Only one of $u, v$ can be in S . Hence, at least one of
$u, v$ is in $V − S$ . So, $V − S$ is a vertex cover.
⇐ Suppose $V − S$ is a vertex cover, and let $u, v ∈ S$ . There
can’t be an edge between $u$ and $v$ (otherwise, that edge wouldn’t
be covered in $V − S$ ). So, S is an independent set. #
But when I construct a Graph as shown bellow:

I supposed the blue nodes to be the Independent Set IS. and after that I picked up an edge $e1= (5,6)$. but I see it does not satisfy the first condition of the proof "...and let $ e = (u, v )$
be some edge. Only one of $u, v$ can be in S ", Where is the mistake?

In your example the blue vertices are an independent set and the remaining ones are vertex cover. Note that a vertex cover can contain an edge, so containing the edge 5-6 is no problem.
