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.