Can I make an assumption about arbitrary numbers in a proof? Assume  $ x_1, x_2... x_{10} $ are different numbers and $ y_1, y_2... y_{10} $ are some arbitrary numbers. Prove that there exists some unique polynomial of degree not exceeding 9 such that:
$P(x_1)=y_1,P(x_2)=y_2...P(x_{10})=y_{10}$
Proof to the problem rests on the assumption that there is some polynomial Q(x) of degree not exceeding 9 such that:
$ Q(x_1)=y_1,Q(x_2)=y_2...Q(x_{10})=y_{10}$. 
Is it a valid assumption in a proof of this sort?

You need to prove two things: that there is such a polynomial and that there is only one.  To prove that there is one, you can show a way to construct it.  Can you find a way to construct a polynomial $Q_1(x)$ of degree not exceeding $9$ such that $Q_1(x_1)=y_1,Q_1(x_2)=0...Q_1(x_{10})=0$?  Now if you construct nine other polynomials with similar conditions and add them together, you get the $Q(x)$ you want.  Linearity is wonderful.  Now you need to prove it unique.  Assume there is another polynomial $R(x)$ of degree at most $9$ that satisfies the requirement.  How many roots does $R(x)-Q(x)$ have?
