Force of decrement question

In the textbook I'm reading, I'm having difficulty understanding how the force of decrement at age x if competing forces are operating and the force of decrement at age x if competing forces are not operating are the same. The text mentions that the reason for this is because the force is an instantaneous rate. The text goes on to say that we can drop the prime notation as a result.



Can someone provide an intuitive explanation as to why the forces are equivalent due to the fact that they're "instantaneous"? To me, this doesn't seem to be a sufficient explanation. I would imagine that regardless of the the instantaneous factor, they would be different, because ultimately, the survival and failure probabilities are different depending on if there are competing risks. And of course, survival/failure probabilities are directly dependent on the force of decrement.



Note: I have a good understanding of competing risks, qx' versus qx, etc. It's just the force of decrement that's causing a problem for me.



Thanks in advance.

Force of decrement question