Lemma 32.16.3. Let $f : X \to S$ and $h : U \to X$ be morphisms of schemes. Assume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & S } \]
where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper.
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