about using Van Kampen theorem

Question

Please see the following link: https://math.stackexchange.com/a/1190643/287172

I saw this question and have understood the answer but, I think in this answer we must prove $ \pi_{1}( Y \cap X_{n+1})= 0 $. Why don't we need this?

Thanks for your help.

Answer 1

That is because by Van Kampen's theorem $\pi_1(Y \cup X_{n+1})$ is a quotient of the free product $\pi_1(Y)*\pi_1(X_{n+1})$ which is trivial since $$\pi_1(X_{n+1})=\pi_{1}(Y)=0\ .$$