I was studying Alexander Holevo's book on **Quantum Systems, Channels and Information**, which is a small but terse introduction to Quantum Information Theory. The theory of Tensor products is used almost everywhere so I decided to study the same topic from different books. There is one property that I find hard to prove and I hope that someone could offer me some assistance in proving it.

N.B: As math stackexchange people may not be comfortable with the physics style notations (a.k.a Dirac's bra and ket notation), I'll use notations from John Conway Functional Analysis. If requested, I'll rephrase the question in Dirac's notation.

Let $\mathcal{H}\_1$ and $\mathcal{H}\_2$ be two finite dimensional Hilbert spaces (over $\mathbb{C}$) with inner products $\langle.\rangle\_1$ and $\langle.\rangle\_2$ respectively.

Let $\{e\_i\}$ and $\{f\_j\}$ be bases for $\mathcal{H}\_1$ and $\mathcal{H}\_2$ respectively.

Let $T \in \mathcal{L}(\mathcal{H}\_1 \otimes\mathcal{H}\_2)$, where $\otimes$ represents the tensor product. Then for any operator $S \in \mathcal{L}(\mathcal{H}\_2)$, show that 
$$Tr\_{\mathcal{H}\_2}(T(I\_{H\_1}\otimes S)) = Tr\_{\mathcal{H}\_2}((I\_{H\_1}\otimes S)T)$$

where $Tr\_{H\_2}(A)$ for any operator $A \in \mathcal{L}(\mathcal{H}\_1 \otimes\mathcal{H}\_2)$, called the *partial trace* of $A$ w.r.t $\mathcal{H}\_2$, is defined as that operator on $\mathcal{H}\_1$ that satisfies the following (this is as per Holevo's book)

$$\langle\phi,Tr\_{H\_2}(A) \psi\rangle\_1 = \sum\_{j}\langle\phi \otimes f\_j,A (\psi \otimes f\_j)\rangle$$
for every $\phi,\psi \in \mathcal{H}\_1$. The inner product on RHS is the inner product for $\mathcal{H}\_1 \otimes \mathcal{H}\_2$ and is given by
$$\langle \phi\_1 \otimes \phi\_2, \psi\_1 \otimes \psi\_2 \rangle = \langle \phi\_1, \psi\_1 \rangle\_1 \langle \phi\_2, \psi\_2 \rangle\_2$$ 
for any $\phi\_i, \psi\_i \in \mathcal{H}\_i;$ $i=1,2$.

It is somewhat similar to showing $Tr(AB)=Tr(BA)$ for the usual trace. However I had a lot of trouble showing this. Also I wanted to know if it suffices to prove the result for $\phi = \psi$ as I think I've managed some kind of proof for that part.