Let $A$ be euclidian ring and $K$ be its field of a fraction.Let $(V,B)$ be a nonzero IPS (inner product space) over $K$. A finitely generated A-submodule $L ⊆ V $is said to be an $A$ -lattice in $V$ if $L$ contains a $K$-basis of $V$ . As we have already observed, $L$ must be $A$-free since it is torsion-free, and it is easy to verify that an $A$-basis for $L$ will automatically be a $K$-basis for $V$.
Attempt: I know that $V$ is finitely generated projective module since $(V, B)$ is inner product space and $B$ is regular bilinear form. Now question is what is the meaning of $L$ contains a $K$-basis of $V$ and how to verify $A$-basis for L will automatically be a $K$-basis for $V$ ; it certainly means we can extend from basis of $L$ to basis of $V$?