We can show that this point exists in the solution as follows:
$\begin{aligned}
&eval[l\star](f) = \sum_i a_i (l\star)^i \\
&= \sum_i a_i \phi(T)^i \\
&\text{Since $\phi$ is ring homomorphism:} \\
&= \sum_i a_i \phi(T^i) \\
&\text{Since $\phi$ is $k$-algebra homomorphism:} \\
&= \phi(\sum_i a_i T^i) \\
&= \phi(f) \\
\text{Since $f \in ker(\phi)$:} \\
&= 0
\end{aligned}$

#### § Consistent and inconsistent system $X$ over ring $L$

Fix a $K$-algebra $L$. The system $X$ is **consistent** over $L$ iff
$Sol(X, L) \neq \emptyset$. the system $X$ over $L$ is **inconsistent** iff
If $Sol(X, L) = \emptyset$.
#### § Geometric Language: Points

Let $K$ be the main ring, $X \equiv (K[T_1, \dots T_n], \mathbf F)$ a system of
equations in $n$ unknowns $T_1, \dots, T_n$.
For any $K$-algebra $L$, we consider the set $Sol(X, L)$ as a collection
of points in $L^n$. These points are solutions to the system $X$.
#### § The points of $K$-algebra

#### § References