Pythagorean Theorem in an Inner Product Space

Question

This is a question about understanding a proof. Here is what my professor did.

However, I do not know how he reached the final conclusion where he states that

$$ \begin{align} \|f\|^2 - 2 \sum_{k=1}^n a_k c_k + \sum_{k=1}^n a_k^2 = \|f\|^2 - \sum_{k=1}^n c_k^2 + \sum_{k=1}^n (c_k - a_k)^2 \end{align} $$

Answer 1

$-c_k^2+(c_k-a_k)^2=-c_k^2+c_k^2-2c_ka_k+a_k^2=-2a_kc_k+a_k^2$.

Answer 2

Observe:

$\displaystyle \sum_{k= 1}^n (c_k - a_k)^2 - \sum_{k = 1}^n c_k^2 = \sum_{k= 1}^n (c_k^2 - 2c_k a_k + a_k^2) - \sum_{k = 1}^n c_k^2 = \sum_{k = 1}^n a_k^2 - 2 \sum_{k = 1}^n a_k c_k. \tag 1$