The following statement is made in the Wikipedia article on small cancellation theory without reference or proof. Can anyone either provide a proof or point me to a reference with a proof?
The statement: "If $R$ and $S$ are finite symmetrized subsets of $F(X)$ with equal normal closures in $F(X)$ such that both presentations $\langle X | R \rangle$ and $\langle X | S \rangle$ satisfy the $C'(1/6)$ condition then $R = S$."
Dehn's algorithm/Greendlinger's lemma give this: By Dehn's algorithm, in a $C'(1/6)$ the shortest trivial words of length greater than $0$ are exactly the shortest relations, so $R$ and $S$ have the same shortest relations. Consider $R_1,S_1$ which are the relation sets with the shortest relations removed.
Claim: $R_1,S_1$ have the same shortest relations. Let $r_0$ be the length of the shortest relations in $R/S$, and let $r_1,s_1$ be the length of the shortest relation in $R_1,S_1$. We can assume without loss of generality that $r_0< s_1 \leq r_1$. Let $w$ be a shortest relation in $S_1$, and apply a step of Dehn's algorithm with respect to the presentation $R$, and note that we can not shorten it with respect to a shortest word in $R$ (that would mean $w$ breaks $C'(1/6)$ in $S$) so we actually use a relation in $R_1$, which means we shorten with a word that is at least as large as $w$. This new reduced word is $v \bar v$ where $\bar v$ is a subword of $w$ with less than half its length, and $v$ is less than half the length of some word in $R_1$.
$|v \bar v| The above argument can be continued for $R_2,S_2$ and so on, till you exhaust $R$ and $S$. A quick sketch of this is you will have $0 Two standard sources which prove Greendlinger's lemma and Dehn's algorithm are Combinatorial Group Theory by Lyndon and Schupp and another standard source is Geometry of Defining Relations in Groups by Olʹshanskii.