I am working my way through a textbook, and it contains the following identity: $$\frac{1}{|C_k|} \sum_{i,i'\in C_k}\sum_{j=1}^p (x_{ij}-x_{i'j})^2 = 2\sum_{i\in C_k}\sum_{j=1}^p(x_{ij}-\bar{x}_{kj})^2$$ where $$\bar{x}_{kj} = \frac{1}{|C_k|}\sum_{i \in C_k}x_{ij}$$
I'm having difficulty proving this to be true. I've tried expanding both sides and transforming one side into the other, but I think I'm having difficulty with some of the notation. Can someone help me construct a proof of this, and maybe explain the notation along the way? I've never seen a summation like the first sigma on the left-hand side of the equation.