<P> First consider the case, where p t − 1 = 0 (\ displaystyle \ textstyle p_ (t - 1) = 0), and note that it implies that q t − 1 = 0 (\ displaystyle \ textstyle q_ (t - 1) = 0) and r t − 1 = 1 (\ displaystyle \ textstyle r_ (t - 1) = 1). Now consider the remaining case, where p t − 1 (\ displaystyle \ textstyle p_ (t - 1)) ≠ 0 (\ displaystyle \ textstyle 0) </P> <Dl> <Dd> 0 = p t − 1 (p t − 1 + 2 q t − 1 + q t − 1 2 / p t − 1 − 1) = q t − 1 2 / p t − 1 − r t − 1 (\ displaystyle (\ begin (aligned) 0& = p_ (t - 1) (p_ (t - 1) + 2q_ (t - 1) + q_ (t - 1) ^ (2) / p_ (t - 1) - 1) \ \ & = q_ (t - 1) ^ (2) / p_ (t - 1) - r_ (t - 1) \ end (aligned))) </Dd> </Dl> <Dd> 0 = p t − 1 (p t − 1 + 2 q t − 1 + q t − 1 2 / p t − 1 − 1) = q t − 1 2 / p t − 1 − r t − 1 (\ displaystyle (\ begin (aligned) 0& = p_ (t - 1) (p_ (t - 1) + 2q_ (t - 1) + q_ (t - 1) ^ (2) / p_ (t - 1) - 1) \ \ & = q_ (t - 1) ^ (2) / p_ (t - 1) - r_ (t - 1) \ end (aligned))) </Dd> <P> Where the final equality holds because the allele proportions must sum to one . In both cases, q t − 1 2 = p t − 1 r t − 1 (\ displaystyle \ textstyle q_ (t - 1) ^ (2) = p_ (t - 1) r_ (t - 1)). It can be shown that the other two equilibrium conditions imply the same equation . Together, the solutions of the three equilibrium equations imply sufficiency of Hardy's condition for equilibrium . Since the condition always holds for the second generation, all succeeding generations have the same proportions . </P>

What are the assumptions of the hardy weinberg law