Higher interior regularity From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post.

THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume $$a^{ij},b^i, c \in C^{m+1}(U) \quad (i,j = 1,\ldots,n) \tag{25}$$ and $$f \in H^m(U). \tag{26}$$ Suppose $u \in H^1(U)$ is a weak solution of the elliptic PDE $$Lu=f \quad \text{in }U.$$
$\qquad$Then $$u \in H_{\text{loc}}^{m+2} (U); \tag{27}$$ and for each $V \subset \subset U$ we have the estimate $$\|u\|_{H^{m+2}(V)} \le C(\|f\|_{H^m(U)}+\|u\|_{L^2(U)}),$$ the constant $C$ depending only on $m,U,V$ and the coefficients of $L$.
Proof. 1. We will establish $\text{(27), (28)}$ by induction on $m$, the case $m=0$ being Theorem 1 above.
$\qquad$2. Assume now assertions $\text{(27)}$ and $\text{(28)}$ are valid for some nonnegative integer $m$ and all open sets $U$, coefficients $a^{ij}, b^i, c$, etc., as above. Suppose then $$a^{ij},b^i,c \in C^{m+2}(U), \tag{29}$$ and $$f \in H^{m+1}(U) \tag{30},$$ and $u \in H^1(U)$ is a weak solution of $Lu=f$ in $U$. By the induction hypothesis, we have $$u \in H_\text{loc}^{m+2}(U) \tag{31},$$ with the estimate $$\| u \|_{H^{m+2}(W)} \le C(\|f\|_{H^m(U)}+\|u\|_{L^2(U)}), \tag{32}$$ for each $W \subset \subset U$ and an appropriate constant $C$, dpending only on $W$, the coefficients of $L$, etc. Fix $V \subset \subset W \subset \subset U$.
$\qquad$3. Now let $\alpha$ be any multiindex with $$|\alpha|=m+1, \tag{33}$$ and choose any test function $\tilde{v} \in C_c^\infty (W)$. Insert $$v :=(-1)^{|\alpha|} D^\alpha \tilde{v}$$ into the identity $B[u,v] =(f,v)_{L^2(U)}$, and perform some integrations by parts, eventually to discover $$B[\tilde{u},\tilde{v}]=(\tilde{f},\tilde{v}) \tag{34}$$ for $$\tilde{u} := D^{\alpha}u \in H^1(W) \tag{35}$$ and $$\tilde{f} := D^\alpha f-\sum_{\substack{\beta \le \alpha \\ \beta \not= \alpha}} {\alpha \choose \beta} \left[-\sum_{i,j=1}^n (D^{\alpha - \beta} a^{ij} D^\beta u_{x_i})_{x_j} + \sum_{i=1}^n D^{\alpha - \beta} b^i D^\beta u_{x_i} + D^{\alpha - \beta} c D^\beta u \right]. \tag{36}$$

My try:
The definition of $B[u,v]$, according to the definition on page 314,
$$B[u,v] := \int_U \sum_{i,j=1}^n a^{ij} u_{x_i} v_{x_j} + \sum_{i=1}^n b^i u_{x_i} v + cuv \, dx$$ and the definition of the inner product $(f,v)$ is given by the textbook as $$f(u,v) = \int_U fv \, dx$$ for $u,v \in H_0^1(U)$ and $f \in L^2(U)$.
So the identity $B[u,v]=(f,v)_{L^2(U)}$ can be expressed as $$\int_U \sum_{i,j=1}^n a^{ij} u_{x_i} v_{x_j} + \sum_{i=1}^n b^i u_{x_i} v + cuv \, dx = \int_U fv \, dx.$$
As per the textbook's instructions, I inserted the identity of $v :=(-1)^{|\alpha|} D^\alpha \tilde{v}$ into the identity and obtained this:
$$\int_U \sum_{i,j=1}^n a^{ij} u_{x_i} (-1)^{|\alpha|} D^\alpha \tilde{v_{x_j}} + \sum_{i=1}^n b^i u_{x_i} [(-1)^{|\alpha|} D^\alpha \tilde{v}] + cu [(-1)^{|\alpha|} D^\alpha \tilde{v}] \, dx = \int_U f [(-1)^{|\alpha|} D^\alpha \tilde{v}] \, dx.$$
I am trying to obtain the result of $$B[\tilde{u},\tilde{v}]=(\tilde{f},\tilde{v}) \tag{34}$$ Am I doing this right so far? Is this where I am ready to perform some integrations by parts?
To show that, you must start with $B[u,v]=(f,v)$, where $v=(-1)^{|\alpha |} D^\alpha \overline{v}.$ So, you need change every partial derivative of $v$ to $u$, by integration by parts.

You will need some theorems in multiindex notation, like Leibniz formula, during of integration by parts and later you will conclud that $B[\overline{u},\overline{v}]=(\overline{f},\overline{v}).$ 

(I'm studying PDE and finished calculation now!)
To show that, you must start with $B[u,v]=(f,v)$, where $v=(-1)^{|\alpha |} D^\alpha \overline{v}.$ So, you need change every partial derivative of $v$ to $u$, by integration by parts.

You will need some theorems in multiindex notation, like Leibniz formula, during of integration by parts and later you will conclud that $B[\overline{u},\overline{v}]=(\overline{f},\overline{v}).$ 

(I'm studying PDE and finished calculation now!)