# Theorem in Dirac's "Principles of quantum mechanics"

I'm pretty new to quantum mechanics and after reading the Susskind's book I dived into the Dirac's one. I've managed to understand until this theorem has been enunciated (in my edition at page 32):

<<There are so many eigenkets of $$\xi$$ that any ket whatever can be expressed as a sum of such eigenkets.>>

I had several doubts about Dirac's procedure therefore I'm trying to ask for someone explining me how have this theorem been proved. For example when Dirac defines $$\chi_r (c_r)$$, he says that it is the number obtained by replacing $$\xi$$ in the algebric expression for $$\chi_r (\xi)$$. This last has been defined as:

$$\chi_r (c_r) = {\phi(\xi) \over {\xi - c_r}}$$

Therefore for the definition of $$\phi (\xi)$$:

$$\chi_r (\xi) = {{(\xi - c_1)(\xi - c_2) \cdots (\xi-c_n)} \over {\xi - c_r}} \Rightarrow \chi_r (c_r) = {{(c_r - c_1)(c_r - c_2) \cdots (c_r-c_n)} \over {c_r - c_r}}$$

but doing this, there has been obtained an expression divided by zero... Could anyone explain me this proof step by step, please?

P.S.: Thank you very much for your time and excuse me for my English, I'm still practising it.

Dirac says that $$\phi(\xi)$$ has the following factorization: $$\phi(\xi) = (\xi -c_1)\dots(\xi -c_n)$$
Hence, the factor $$(\xi -c_r)$$, with $$1 \leq r \leq n$$, divides $$\phi$$. Then, he goes on to prove that the $$c_i$$ are all different. This in turn implies that $$\chi_r(c_r) \neq 0$$.