# How does the following term happen when using the Wald Test on two-sample binomials?

I'm having trouble understanding one of the steps of the Wald test used in two-sample binomial proportions. More specifically, when $$X_1, ..., X_{n_1}$$ are iid Binomial$$(1, p_1$$), $$Y_1, ..., Y_{n_2}$$ are iid Binomial$$(1, p_2$$), with these two samples mutually independent, and the hypothesis being $$H_0: p_1=p_2$$ versus $$H_1:p_1\neq p_2$$

Supposing $$n_1/n \to\lambda_1 \in(0,1)$$, and $$n_2/n \to\lambda_2 \in(0,1)$$, as $$n=n_1+n_2\to\infty$$, using the central limit theorem and Slutsky's theorem the following apparently can be found: $$\sqrt n[(\hat p_1-\hat p_2)-(p_1-p_2)]$$ converges in distribution to $$N(0,\frac{1}{\lambda_1}p_1(1-p_1)+\frac{1}{\lambda_2}p_2(1-p_2))$$

Could someone give me some insight or a basic proofing of how the final term can be found? Thank you in advance.