In Example BP we claimed that \begin{align*} B&=\set{1,\,x,\,x^2,\,x^3,\,\ldots,\,x^n}\\ C&=\set{1,\,1+x,\,1+x+x^2,\,1+x+x^2+x^3,\,\ldots,\,1+x+x^2+x^3+\cdots+x^n}. \end{align*} were both bases for $P_n$ (Example VSP). Suppose we had first verified that $B$ was a basis, so we would then know that $\dimension{P_n}=n+1$. The size of $C$ is $n+1$, the right size to be a basis. We could then verify that $C$ is linearly independent. We would not have to make any special efforts to prove that $C$ spans $P_n$, since Theorem G would allow us to conclude this property of $C$ directly. Then we would be able to say that $C$ is a basis of $P_n$ also.