A source coding system with side information at the decoder is a typical multiterminal source coding system where output sequences of two sources are independently encoded, but a decoder recovers only one output sequence from two codewords. Since Wyner, Ahlswede and Korner independently investigated this system, we call it as the WAK coding system. This paper investigates the e-achievable rate region of the WAK coding system which allows the probability of error within a fixed tolerance e(∈ (0, 1)), and clarifies the e-achievable rate region for correlated general sources in terms of the smooth max-entropy and the smooth max Renyi divergence. To this end, we show a new one-shot converse theorem for the WAK coding system, and a one-shot covering lemma which is a refined version of Warsi's result. Then, combining these results, we clarify the e-achievable rate region of the WAK coding system.