Proving $S=SRimplies S+phi$

Let $Lsubseteq Sigma^*$ such that ${epsilon}notin L$. Then for any $Ssubseteq Sigma^*,
S=SLimplies S=phi$

So we suppose $S=SL$ and $Snephi$. Then $exists win S$ such that $0le|w|le |v|$ for some $vin S$. Now $|w|ne 0$, since $|w|=0implies{epsilon}=win S=SLimplies {epsilon}=xy$ for some $xin S$ and $yin Limplies x=y={epsilon}in L$ which contradicts that ${epsilon}notin L$.
Thus $wne{epsilon}$ and ${epsilon}notin S$. So we have
w in S &implies win SL\
&implies w=sl;text{for some}; sin S; text{and}; lin L\
&implies |w|=|s|+|l|

Here I’m stuck now. I think if I can show that $|l|=0$, then we can arrive at a contradiction.

Your help is highly appreciated.