Struggling with a simple step from the Variance Theorem

[lsiqueirap]

I'm studying statiscs by George Casella's text book and I came across the following theorem:


How come this cross term can be 0?

 

[lev888]

x₁ + ... + x_n - n*average = 0

 

[lsiqueirap]

Thanks a lot. Struggling with blazingly a simple thing is awful.