Given the mutually independent rv $X\_1,X\_2,Y\_1,Y\_2$, I want to show that $(X\_1+Y\_1),(X\_2+Y\_2)$ are independent as well. Is it sufficient to show that $$E((X\_1+Y\_1)(X\_2+Y\_2))=E(X\_1+Y\_1)E(X\_2+Y\_2)$$?