Linear subspace/Intersection/Dimension estimate/Fact

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space of dimension ${}n$. Let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces of dimension ${}\dim _{K}{\left(U_{1}\right)}=n-k_{1}$ and ${}\dim _{K}{\left(U_{2}\right)}=n-k_{2}$.

Then

${}\dim _{K}{\left(U_{1}\cap U_{2}\right)}\geq n-k_{1}-k_{2}\,.$

Proof, Write another proof