# T, U, W are vector subspaces of V. If $T \subset W, U \cap W = U \cap T$ and $U + W = U + T$ then $T=W$.

I have problems proving the following:

Let V be a vector space over a field K. Let T, U, W be vector subspaces of V. If $T \subset W$, $U \cap W = U \cap T$ and $U + W = U + T$, then $T=W$.

Proving $T=W$ can be reduced to proving $W \subset T$, so I have to prove: $x \in W \Rightarrow x \in T$

I started with:$$x \in W \Rightarrow x \in U + W \Rightarrow x \in U + T \Rightarrow x \in \{a + b|a \in U \wedge b \in T\}$$ Now there has to be a $b \in T$ such that $b = x$, but I have no idea how to continue from here.

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@amWhy : $T \subseteq W$ is assumed. –  Patrick Da Silva Nov 17 '12 at 20:46

Let $x \in W$, then $x \in U + W = U + T$, hence $x = u + t$ for some $u\in U$, $t \in T$. Then $x-t = u \in U$ and in $W$ (since $T \subseteq W$), so $u \in U \cap W = U \cap T$. Hence $u \in T$ and so $x = u+t \in T$.