Show that the $A/B$ is a normal subgroup of $G/H.$

Question

Goal: To show that $A_{i}/A_{i+1}$ is a normal subgroup of $B_i/B_{i+1}.$

We are given that

1. $B_{i} \triangleright B_{i+1}.$
2. $B_{i} \triangleright A_i.$
3. $A_i \triangleright A_{i+1}.$
4. $A_{i}\subset B_{i}$
5. $A_i=B_1\cap K_1\cap B_i$, where $K_1$ is another normal subgroup of $G.$

where $G\triangleright H$ means that $H$ is a normal subgroup of $G.$

I have taken the projection $\pi:B_i\to B_i/B_{i+1}.$ Then $$\ker(\pi_{|A_i}) = A_i\cap B_{i+1} = B_1\cap K_1\cap B_i \cap B_{i+1} = A_{i+1}.$$ So $$A_i/A_{i+1}\cong \text{image}(\pi_{|A_i})$$ since the $ \text{image}(\pi_{|A_i})$ is normal in $B_{i}/B_{i+1}$ we have that $A_i/A_{i+1}$ is a normal subgroup in $B_{i}/B_{i+1}.$ Is this correct reasoning?

This is related to the question I previously asked here.