Representation of nullhomologous loop on compact surface as a product of commutators. Why this sentence is true?:
Assume that $M$ is compact surface and $f: S^1 \to M$ is nullhomologous and without selfintersections. Letting $g$ be the genus and $b$ the number of boundary components of $M$, it follows that there is a generating set $S=\{\alpha_1, \beta_1, ..., \alpha_g, \beta_g, x_1, ..., x_b\}$ such that:
$$ \pi_1(M, *) = \langle a_1, b_1, \ldots , a_g, b_g, x_1, \ldots , x_b \mid [a_1, b_1]\cdots [a_g, b_g]= x_1\cdots x_b \rangle . $$ 
and such that:
$f$ is homotopic to $[\alpha_1, \beta_1] \cdots [\alpha_{g'}, \beta_{g'}]$ for some $g'<g$.
I know Hurewicz teorem, so I know that $f$ belongs to comutant of fundamental group, but why representation of $f$ is such as above? In particular why in this representation doesn't $x_i$ appear?
Could anybody help me or show me the book,where I can read about it in a algebraic topology way (I'm not familiar with differential topology)?
[Edit]: I forgot to write, that $f$ is without selfintersections 

Updated after OP's edit:
Here's the idea: A simple closed curve $\gamma$ in an orientable genus $g$ surface $M$ is nullhomologous if and only if $M \setminus \gamma$ consists of two connected components, one of which is a surface $N$ with $\partial N = \gamma$. Since $\gamma$ is the one and only boundary component of $N$, all of the boundary components corresponding to $x_1,\ldots,x_b$ lie in the second connected component (which has a total of $b+1$ boundary components, when you include its copy of $\gamma$). Since $N$ is an orientable surface of genus $g'\leq g$, we can choose standard generators $\alpha_1,\beta_1,\ldots,\alpha_{g'},\beta_{g'}$ of $\pi_1(N)$ such that $\gamma$, the boundary of $N$, has homotopy class $[\alpha_1,\beta_1]\cdots[\alpha_{g'},\beta_{g'}]$. Then $\alpha_i,\beta_i$ are also generators in $\pi_1(M)$. The other generators $\alpha_{g'+1},\beta_{g'+1},\ldots,\alpha_g,\beta_g,x_1,\ldots,x_b$ come from the second connected component.
Remark. Note that we must allow $g'\leq g$, not "$g'<g$". Suppose $M$ has $b$ punctures and $\gamma$ is a small loop around these punctures. On one side, $\gamma$ bounds a disk with $b$ punctures. On the other side, $\gamma$ bounds a genus $g$ surface with a disk removed. Thus $\gamma$ is nullhomologous, but in terms of the standard generators its homotopy class is $[\alpha_1,\beta_1]\cdots[\alpha_g,\beta_g]$.
