I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs. 

Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a vector bundle of rank $r$ and degree $d\geq0$ over $X,$  then $s:=h^0(X,E)=d$ if $d>0$ and $s=0$ or $1$ if $d=0.$ 

For the proof, if $d\geq r,$ choose a maximal splitting $E=(L_1 ,...,L_r)$ with each $L_i>1$ (where 1 denotes the trivial line bundle). But in Lemma 11 he has the equation 

$d=deg(E)=\sum^{r}_{1} deg(L_i).$ If each $L_i>1$ doesn't that mean $deg(L_i )\geq2$ so $deg(E)\geq2r?$ But we are only assuming $d\geq r.$

 Anyways, If $deg(L_i)\geq 2 $ then it's clear that $H^1 (X,L_i)=0$. Is that how he claimed that $H^1 (X,L_i)=0$ in the next line or is he using something else?