$U = \{(a, b, c)\ :\ a = b = c\}$ e $W = \{(0, b, c)\}$, $\mathbb{R}^3 = U \oplus W$.

$\{(0, 0, 0)\} = U \cap W\ {\large (I)}$

Seja $(a, b, c)$ um vetor do $\mathbb{R}^3$:

$(a, b, c) = (a, a, a) + (0, b - a, c - a)\ \Rightarrow\ \mathbb{R}^3 = U + W\ {\large (II)}$.

${\large (I)}\ \wedge\ {\large (II)}\ \Rightarrow\ \mathbb{R}^3 = U \oplus W$

Quod Erat Demonstrandum.