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.