# algebraic topology – property on 2 fibrations?

Let $$X xrightarrow {f} Y xrightarrow {g} Z$$ to be a diagram in the category of topological spaces. Yes $$g circ f$$ and one of $$f, g$$ are fibrations, can we conclude that the other is also?

In this question it is stated that if $$f$$ is surjective and $$g circ f$$ and $$f$$ are fibrations then $$g$$ is a fibration. However, no proof is given. Is it true? How can I prove it? I understand that the main difficulty is that an arbitrary card $$W to Y$$ can not be lifted along $$f$$ to a map $$W to X$$. I'm not sure this lifting property is true …

The motivation for this problem is that I want to show that $$mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n}$$ and $$mathbb {CP} ^ {2n + 1} to mathbb {HP} ^ n$$ are fibrations. I know that there are fibrations (actually bundles of fibers) $$mathbb {S} ^ {2n + 1} to mathbb {RP} ^ {2n + 1}$$ and $$mathbb {S} ^ {2n + 1} to mathbb {CP} ^ n$$ and the factoring of these two cards gives the map $$mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n}$$. I wonder if we could use the property above to infer directly that $$mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n}$$ is a fibration.