I have this question from Kratzer and Thévenaz – Type of motor surgery of the exercises and the activities of the groups of people:

On page 89, they explain that

Yes $ Y = Y_1 cup Y_2 $ They are such that $ X = Y_1 cap Y_2 $ and if each one $ Y_i $ it's a cone about $ X $, so $ Y $ is the suspension of $ X $

$ def abs # 1 { lvert # 1 rvert} $But on page 90, in the test of the **Proposition 2.5**, they use this in a way that I do not understand. They prove that $ abs X simeq operatorname C abs E $ Y $ abs Y simeq operatorname C abs F $, but those are not cones about $ abs {X cap Y} $, then, why can I continue the proof of this proposition by saying that these implications make $ abs G simeq sum ( abs {X cap Y}) $?

My concept of a "cone on $ abs {X cap Y} $" is:

$$ abs {X cap Y} times [0,1]/ (a, i) thicksim (b, 1) $$

with $ a, b in X cap Y $. Am I misunderstanding what a cone is? $ abs {X cap Y} $?