For the $n$-th order nonlinear differential equation, ${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k\,+\,j)$-point boundary conditions for $1\,\le \,j\,\le \,n\,-\,1$ and $1\,\le \,k\,\le \,n\,-\,j$. We define $(k;\,j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n\,-\,{{j}_{0}};\,{{j}_{0}})$-point unique solvability implies $(k;\,j)$-point unique solvability for $1\,\le \,j\,\le \,{{j}_{0}}$, and $1\,\le \,k\,\le \,n\,-\,j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\,\le \,k\,\le \,n\,-\,1$.