Here is the relation I am trying to understand
Let $C$ be a circuit of a matroid $M.$ For e in $E(M)$:
If $e \in C,$ then $e$ is a loop of $M,$ or $C - e$ is a circuit of $M/e$ ( M contract e).
Does anyone has a justification for this idea? it is used a lot but I do not know why this is true.
Well, either $C = \{e\}$ or not, if it is, then $C$ is a loop. If not, then $|C|>1$ and in $M/e$ the independents are independents of $M$ such that when added $e$ they remain independent. In this way, $C$ is not independent, so its dependent. Furthermore, because $C$ is circuit if $f\in C\setminus \{e\}$ then $C\setminus \{f\}$ is independent and so $C\setminus \{f,e\}$ is independent in $M/e$ for every $f$, hence $C\setminus \{e\}$ is minimal dependent, which is the definition of being a circuit.