I can't see why the following equalities are valid:
Let $X$ be an inner product space with an othonormal system $e_n, n\in\mathbb{N}$ and $x\in X$.
Define $y_n=\sum_{i=1}^n\langle x;e_i\rangle e_i$, in wich case $$\underline{\Vert y_n\Vert^2=\langle x;y_n\rangle=\langle y_n;x\rangle }=\sum_{i=1}^n\vert\langle x;e_i\rangle\vert^2$$
I get the last part, however I don't see the underlined part.
$$ \langle y_n;y_n \rangle = \langle \sum_{i=1}^n\langle x;e_i \rangle e_i,\sum_{j=1}^n\langle x;e_j \rangle e_j\rangle =\sum_{i=1}^n\sum_{j=1}^n\langle x;e_i \rangle\overline{\langle x;e_j \rangle} \underbrace{\langle e_i;e_j \rangle}_{=\delta_{i,j}} = \sum_{k=1}^n \langle x;e_k\rangle\overline{\langle x;e_k\rangle} = \langle x; \sum_{k=1}^n \langle x;e_k\rangle e_k\rangle = \langle x;y_n\rangle$$
where I used the fact that the basis $e_1,\ldots,e_n$ is orthonormal and thus $\langle e_i;e_j\rangle = \delta_{i,j}$ where $\delta_{i,j}$ is the Kronecker delta. Finally since $\| y_n\|$ is real, it holds $$\langle x;y_n\rangle=\| y_n\|= \overline{\| y_n\|} = \overline{\langle x;y_n\rangle} = \langle y_n; x\rangle.$$