Independence of Sample Mean and sample Variance of Normal distribution.

<Independence of Sample Mean and Variance of Normal distribution>

I want to prove that when random samples X1,X2,...,Xn independent and identically follows Normal distribution with mean μ and variance σ^2, the sample mean and the sample variance of the samples are independent. 

This is regarded as a crucial fact when proving one of the properties of T distribution. Which is when x1,x2,x3,...,xn are from the normal distribution with mean μ and variance σ^2, n^0.5*(sample mean-μ)/sample variance follows T distribution with n-1 degrees of freedom. Let me firstly prove the Independence and show how it works importantly in T distribution.