## $X,Y$ spaces such that are cones over $X\cap Y$ generate $X\cup Y=\sum (X\cap Y)$

I have this question from this Kratzer and Thevenaz article :

In page 89, they explain that

If $$Y=Y_1\cup Y_2$$ are such that $$X=Y_1\cap Y_2$$ and if each $$Y_i$$ is a cone over $$X$$, then $$Y$$ is the suspension of $$X$$

But in page 90, in the proof of the Proposition 2.5, they use this in a way I don’t understand. They prove that $$\mid X\mid \simeq C\mid E\mid$$ and $$\mid Y\mid \simeq C\mid F\mid$$, but those aren’t cones over $$\mid X\cap Y\mid$$, so why I can continue the proof of this proposition saying that this implications make $$\mid G\mid \simeq \sum(\mid X\cap Y\mid)$$ ?

My concept of a “cone over $$\mid X\cap Y\mid$$” is:

$$\mid X\cap Y\mid\times [0,1]/ (a,i)\thicksim(b,1)$$

with $$a,b\in X\cap Y$$.

Am I misunderstanding what is a cone over $$\mid X\cap Y\mid$$?