设\(\xi_1,\cdots ,\xi_r\)是\(V_1\bigcap V_2\)的基。由于\(\dim V_1=\dim V_2\)，所以将其扩充为\(V_1\)的基
\begin{equation*}
\xi_1,\cdots ,\xi_r,\alpha_1,\cdots ,\alpha_s,
\end{equation*}
也可将其扩充为\(V_2\)的基
\begin{equation*}
\xi_1,\cdots ,\xi_r,\beta_1,\cdots ,\beta_s.
\end{equation*}
此时，\(\xi_1,\cdots ,\xi_r,\alpha_1,\cdots ,\alpha_s,\beta_1,\cdots ,\beta_s\)是子空间\(V_1+V_2\)的基。因此，可将其扩充为\(V\)的基\(\xi_1,\cdots ,\xi_r,\alpha_1,\cdots ,\alpha_s,\beta_1,\cdots ,\beta_s,\gamma_1,\cdots ,\gamma_t\)。令
\begin{equation*}
W=\langle \alpha_1+\beta_1,\cdots ,\alpha_s+\beta_s,\gamma_1,\cdots ,\gamma_t\rangle,
\end{equation*}
我们断言，\(V=V_1\oplus W=V_2\oplus W\)。事实上，对任意\(\alpha\in V_1\bigcap W\)，有
\begin{equation*}
\alpha=\sum\limits_{i=1}^rk_i\xi_i+\sum\limits_{j=1}^sl_j\alpha_j=\sum\limits_{j=1}^sp_j(\alpha_j+\beta_j)+\sum\limits_{m=1}^tq_m\gamma_m,
\end{equation*}
则
\begin{equation*}
\sum\limits_{i=1}^rk_i\xi_i+\sum\limits_{j=1}^s(l_j-p_j)\alpha_j-\sum\limits_{j=1}^sp_j\beta_j-\sum\limits_{m=1}^tq_m\gamma_m=0.
\end{equation*}
由\(\xi_1,\cdots ,\xi_r,\alpha_1,\cdots ,\alpha_s,\beta_1,\cdots ,\beta_s,\gamma_1,\cdots ,\gamma_t\)线性无关可知：
\begin{equation*}
k_1=\cdots=k_r=l_1=\cdots=k_s=p_1=\cdots=p_s=q_1=\cdots=q_t=0,
\end{equation*}
则\(\alpha=0\)，故\(V_1\bigcap W=0\)。对任意\(\beta\in V\)，有
\begin{equation*}
\beta=\sum\limits_{i=1}^ra_i\xi_i+\sum\limits_{j=1}^sb_j\alpha_j+\sum\limits_{j=1}^sc_j\beta_j+\sum\limits_{k=1}^td_k\gamma_k,
\end{equation*}
即
\begin{equation*}
\beta=\left[\sum\limits_{i=1}^ra_i\xi_i+\sum\limits_{j=1}^s(b_j-c_j)\alpha_j\right]+\left[\sum\limits_{j=1}^sc_j(\alpha_j+\beta_j)+\sum\limits_{k=1}^td_k\gamma_k\right]\in V_1+W,
\end{equation*}
所以\(V=V_1+W\)，从而\(V=V_1\oplus W\)。同理可证，\(V=V_2\oplus W\)。
