Deterministic context-free languages are unambiguous

Choose a DPDA $M$ recognizing $L$ by [L1].

Form an equivalent DPDA $\widehat{M}$ that records whether the endmarker has been consumed. Before $⊣$ it simulates $M$ in a pre-endmarker copy of the state set; a transition consuming $⊣$ enters the post-endmarker copy; and all subsequent $\epsilon$-moves remain in that copy. Declare a post-endmarker copy accepting exactly when its underlying state lies in $F$, and delete every outgoing move from those accepting copies. This preserves determinism by [L1]: it only duplicates states and removes moves.

The machine $\widehat{M}$ recognizes the same language as $M$. Indeed, an accepting computation of $M$ has consumed the marked input and hence reaches a post-endmarker accepting copy in $\widehat{M}$; conversely, every accepting copy projects to an accepting configuration of $M$. Halting at the first such copy does not change whether that configuration is reachable.

Every accepting computation of $\widehat{M}$ is maximal because its accepting states have no outgoing moves. By [L2], there is therefore at most one accepting computation on each marked input.

Applying [L3] to $\widehat{M}$ yields an unambiguous context-free grammar generating its language, which is $L$ by step 3.1. Therefore every deterministic context-free language is unambiguous.