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2926
```
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14.\begin{window}[1,l,{\includegraphics[width=22mm]{14.png}},{}]
{\heiti 解:}由13题(1)结论知,$(W\bigcap V_1\big)+\big(W\bigcap V_2\big)\subseteq W\bigcap\big(V_1+V_2\big)=W$,并且的确真不相等的例子,例如在几何空间中,设$V_1,V_2,W$是过原点的三个平面,且它们相交于同一条直线L,由于$V_1+V_2=V$,因此,$W\subseteq V_1+V_2. $而$(W\bigcap V_1\big)+\big(W\bigcap V_2\big)=L$,而$W\nsupseteqq L. $如果$V_1\subseteq W$,那么结论就成立了.\\理由如下:任取$\alpha\in W$,由于$W\subseteq V_1+V_2$,因此$\alpha\in V_1+V_2$,从而有$\alpha=\alpha_1+\alpha_2,\alpha_1\in V_1, \alpha_2\in V_2.$由于$V_1\subseteq W$,因此$\alpha_1\in W$,从而$\alpha_2=\alpha+\alpha_1\in W$,于是$\alpha _2\in V_2\bigcap W. $因此得出$\quad \alpha=\alpha_1+\alpha_2\in \big(W\bigcap V_1\big)+\big(W\bigcap V_2\big)$,因此
$W\subseteq\big(W\bigcap V_1\big)+\big(W\bigcap V_2\big)$,又由于$\big(W\bigcap V_1\big)+\big(W\bigcap V_2\big)\subseteq W$,所以$W=\big(W\bigcap V_1\big)+\big(W\bigcap V_2\big). $
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```
![](https://pics.latexstudio.net/data/images/201912/31b9d59718fe063.png)