The Book is Bosch Linear Algebra.

I have difficulties to understand how one deduces this statement from prior statements proved earlier:

Statement already proved:

**Definition 1.**

Let $U\_1,....,U\_r$ be linear subspaces of a $KVR$ $V$. The sum is defined as 

$$\sum\_{i=1}^{r}U\_i=\{\sum\_{i=1}^{r}b\_i;b\_i\in U\_i, i=1,...,r \}$$

**Theorem 2**Let $U=\sum\_{i=1}^{r}U\_i$ be a sum of subspaces of a $KVR$ $V$, the following statements are equivalent:

(i) If $b$ is written as $b=\sum\_{i=1}^{r}$ then the $b\_i$ are unique. (ii) If $\sum\_{i=1}^{r}b\_i=0$ then $b\_i=0$ for all $i$ (iii) $U\_p\cap\sum\_{i\neq p}U\_i=0$ for all $p=1,...,r$.

**Definition 3** If the sum $U=\sum U\_i$ fullfills one of the conditions above in **Theorem 2** the sum is called direct.

**Theorem 4**Let $V$ be a $KVR$ $V$ and $U$ a subspace then there exissts a subspace $U'$ with $V=U+U'$ such that the sum is direct and for all such $U'$ the dimension formula:

$$\dim V= \dim U +\dim U'$$ holds

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> **Theorem 5** $\dim (U+W)= \dim U + \dim W -\dim (U\cap W)$
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> 
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There is a question about the proof, up to a certain point I understand it:

We suppose that $U$ and $U'$ are of finite dimension subspaces of a vectorspace in $V$.

$U+U'$ thus has a finite generating system and thus a finite basis. One chooses in accordance to Theorem 4 a complement $W$ of $U\cap U'$ and $W'$ of $U\cap U'$ such that 

$U=(U\cap U') + W$ where the sum is direct and $U'=(U \cap U') + W'$ where the sum is also direct.

Then 

$U+U'=(U\cap U')+W+W'$ and the sum (on the RHS) is also direct.

Let $a+b+b'=0$ $a\in$ intersection $b\in W$ and $b'\in W'$ then 

$b=-(a+b')\in (U\cap U')+W'=U'$ and because of $b\in W\subset U$. $b\in (U\cap 
 U')\cap W=0$

analogously $b'=0$ then also $a=0$. Therefore the sum is direct

*And because of Theorem 4 we now can say*

$\dim(U+U')=\dim(U\cap U') + \dim W + \dim W'$

This is a sum with $3$ summands, if it would be a sum with two summands then I would understand it. But when it is three sumands I don't know how you can **directly** deduce from Theorem 4 this dimension formula. You can prove it by induction but the author says this can be directly deduced from theorem $4$ and didn't even mention induction. 

I know there are many questions about this formula but I want to understand this specific proof of this formula.