if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is
member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of
$V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less
than and equal to $n$. Let $H$ be a subspace spanned by $B_m$ and $u$.
Moreover, let $K$ be the subspace spanned by $B_m$ and $v$. Prove that if
$v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member
of $K$.