combination of normal distribution samples According to sums of Independent Normal Random Variables 

Let's say we have two samples. Sample 1 follows a N(u1, var1) and Sample 2 follows a N(u2, var2), 
Case 1:
I take one subject from each sample, those two subjects will follow a N(u1+u2,var1+var2), easy to be proved by using moment-generating function of a linear combination of independent random variables.
Case 2:
I have a different situation now, let's say I mixed two samples first, then  I take one subject in the mixed sample, it's intuitive to consider it still follow a N(u1+u2,var1+var2). how can I prove it?
 A: There is a difference between summing random variables and mixing them.  In your post your mixing random variables which leads to a normal mixture distribution.
To see the difference, let 
$$
X_1,X_2,...,X_n \stackrel{iid}{\sim} N(\mu_1,\sigma_1^2)
$$
and
$$
Y_1,Y_2,...,Y_n \stackrel{iid}{\sim} N(\mu_2,\sigma_2^2)
$$
where $X_i$ and $Y_i$ are independent for all $i \in \{1,...,n\}$.
If you created a new variable $Z_1,Z_2,...,Z_n$ where $Z_i = X_i+Y_i$, then $Z_i \stackrel{iid}{\sim} N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$.  However, this is not what you said in your post.  What you said was that you "mixed two samples".
Suppose that you did mix these samples and the observations where unlabeled (meaning that you don't know which original sample each observation came from).  Denote the mixed sample as $A_1, A_2,...,A_{2n}$.  For each $i$ there is a 50% chance that $A_i$ came from sample 1 and a 50% chance it came from sample 2. Thus
$$
A_i \stackrel{iid}{\sim} .5 \times \phi(A_i|\mu_1,\sigma_1^2) +.5 \times \phi(A_i|\mu_2,\sigma_2^2)
$$
where $\phi$ is the normal probability density function.  The above distribution is called a mixture and it differs greatly from the normal distribution you proposed in your post example.  The below plot is an example of a mixture between two normals a $N(-3,1)$ and $N(,2,0.5^2)$ compared to the distirubtion of their sum.   

