Lemma 1 Groups
Let $(G, \circ)$ be a group. For $a, b, c \in G$, it always follows from $a \circ c = b \circ c$ that $a = b$, and from $c \circ a = c \circ b$ it also always follows that $a = b$.
Beweis
Combine $a \circ c = b \circ c$ from the right with $c^{-1}$ and obtain $a \circ c \circ c^{-1} = b \circ c \circ c^{-1} \Leftrightarrow a \circ e = b \circ e \Leftrightarrow a = b$.
Combine $c \circ a = c \circ b$ from the left with the inverse element $c^{-1}$ of $c$ and obtain
\begin{align*}
c^{-1} \circ c \circ a = c^{-1} \circ c \circ b \Leftrightarrow e \circ a = e \circ b \Leftrightarrow a = b
\end{align*}.