I have this question from Kratzer and Thevenaz – Homopopy type of lattices and lattices of a finite group:

On page 89, they explain that

Yes $ Y = Y_1 cup Y_2 $ are such that $ X = Y_1 cap Y_2 $ and if everyone $ Y_i $ is a cone on $ X $then $ Y $ is the suspension of $ X $

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

My concept of "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 $. Do I misunderstand what is a cone? $ abs {X cap Y} $?