Help with this demostration of spaces

Question

Let $U, W, S$ dimensional subspaces of a finite space such that $V=U+W+S$. Prove that $V = U\oplus W\oplus S$ if and only if $\dim (V) = \dim (U) + \dim (W) + \dim (S)$.

Answer 1

$\Longrightarrow)$ Suppose that $V=U \oplus W \oplus S$.

Then, \begin{split} dim(V) &= dim((U \oplus W) \oplus S) \\ &= dim(U \oplus W) + dim(S) - dim((U \oplus W) \cap S). \end{split}

As $$(U \oplus W) \cap S = \{0\}$$ (because we have a direct sum), we have $$dim(V) = dim(U \oplus W) + dim(S).$$

Again, $$dim(U \oplus W) = dim(U) + dim(W).$$

Then, $$dim(V) = dim(U) + dim(W) + dim(S).$$

$\Longleftarrow)$ Suppose that $dim(V)=dim(U)+dim(W)+dim(S)$.

We have \begin{split} dim(V) &= dim(U+W+S) \\ &= dim((U+W)+S) \\ &= dim(U+W)+dim(S)-dim((U+W)\cap S) \\ &= dim(U)+dim(W)-dim(U \cap W)+dim(S)-dim((U+W) \cap S). \end{split}

Then, $$0=dim(U \cap W)+dim((U+W) \cap S).$$

So $$dim(U \cap W) = 0$$ and $$dim((U+W) \cap S)=0.$$

Then, \begin{split} V = (U + W) \oplus S &= (U \oplus W) \oplus S \\ &= U \oplus W \oplus S. \end{split}