Let $r$ be a positive integer such that $r([L]-1)=0$ in $\widetilde{KO}(\mathbb{R}P^{b-1})$, where $L$ is a line bundle. This implies that $rL\oplus \underline{s}\cong \underline{r}\oplus\underline{s}$ for some $s\geq0$.
I am confused with one operation using above isomorphism: $$\Sigma^{-r-s}\text{Th}\left( (a+1)L\oplus\underline{s+r} \right)\simeq \Sigma^{-r-s}\text{Th}\left( (a+1+r)L\oplus\underline{s} \right)$$
So specially I am looking for explanation of how $r$ from $(a+1)L\oplus\underline{s+r}$ goes to $(a+1+r)L\oplus\underline{s}$ due to assumed iso?
Using the fact that $\underline{r}\oplus\underline{s} \cong \underline{r + s}$ we have $$(a + 1 + r)L \oplus \underline{s} \cong (a + 1)L \oplus (rL \oplus \underline{s}) \cong (a + 1)L \oplus (\underline{r} \oplus \underline{s}) \cong (a + 1)L \oplus \underline{r + s}.$$