Every Picard–Lindelöf initial value problem has one maximal solution on an open interval

Take the union $I$ of the domains of all solutions through the initial point; [L1] makes the family nonempty, and the union of intervals containing $t_0$ is an interval. For $t\in I$, choose one such solution $x:J\to {\mathbb{R}}^n$ with $t\in J$. By [L1] at $(t,x(t))$, there is a local solution $y$ on an open interval $V$ about $t$. By [L2], $x$ and $y$ agree on $J\cap V$, so their union is a solution on $J\cup V$ through the original initial point. Hence $V\subseteq I$, proving that $I$ is open.

By [L2], all values assigned at a time in $I$ agree, so their pointwise union is a well-defined solution; every other solution is its restriction, which proves maximality, and the same property forces uniqueness of the maximal solution.