Incremental RC4 algorithm?
If I understood you correctly, the protocol you are analysing doesn't reinitialize the encryption before encrypting, so it seems like it is encrypting everything it send yet and sends only the part that "is new". 

To answer your question, let's take a little excourse into stream ciphers (as RC4). These ciphers basicly generate a stream of random numbers which are then xored with the plaintext. To decrypt, you have to generate the key stream again and then xor it with the ciphertext.

$C = P \oplus K$

$P = C \oplus K$

($C$ is the ciphertext, $P$ is the plaintext, $K$ is the keystream and $\oplus$ is XOR)

One property of such ciphers is that to en- or decrypt one section is that you just need the related section of the keystream. So, in your protocol, following will likely happen:

Client-Side:


Calculates $K_{0}$
Sends $C_0 = P_0 \oplus K_0$
Calculates $K_1$
Sends $C_1 = P_1 \oplus K_1$


Note that the client doesn't need $K_0$ nor $P_0$ to encrypt $P_1$.

Server-Side:


Calculates $K_0$
Recieves $C_0$ and decrypts it using $P_0 = C_0 \oplus K_0$
Calculates $K_1$
Recieves $C_1$ and decrypts it using $P_1 = C_1 \oplus K_1$, the server doesn't need $K_0$ nor $P_0$ to decrypt $C_1$, so it doesn't have to do the operations you did while reverse engineering.