Linear subspace/Sum and intersection/Dimension/Fact

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces.

Then

$\dim _{K}{\left(U_{1}\right)}+\dim _{K}{\left(U_{2}\right)}=\dim _{K}{\left(U_{1}\cap U_{2}\right)}+\dim _{K}{\left(U_{1}+U_{2}\right)}\,.$

Proof, Write another proof