Quantile(X + constant) = Quantile(X) + constant?

Question

I would like to 'prove' that
$$q_{\alpha}(X + c) = q_{\alpha}(X) + c $$ For c $\in \Bbb{R}$, $X$ a random variable, and $q_{\alpha}$ the quantile of order ${\alpha}$. I would actually like to prove this for conditional quantile (but I think it does not really impact the proof).

I saw the 'equivariance of quantile under monotone transformation' property, which is a way stronger property (and I did not find any proof for this either anyway), so I think that $q_{\alpha}(X + c) = q_{\alpha}(X) + c $ is true (or maybe there are some particular cases in which it does not hold? I cannot think of any). The only problem is I don't know how to prove it.
To me it seems quite intuitive, but I'm not sure I am allowed to write this without proving it (and there might be some special case I'm not thinking of).

I would also appreciate if anyone got a link to a formal proof of the equivariance of quantile under monotone transformation.

Thanks in advance.

Answer 1

This is based on $$ P(X+a > t) = P(X > t - a) $$ connected to the definition of $q_\alpha$.