Let's suppose we have 3 independent Gaussian random variables with means m_x, m_y, m_z (they can be different) and variances sigma^2 (the same for all):
X ~ N(m_x, sigma^2)
Y ~ N(m_y, sigma^2)
Z ~ N(m_z, sigma^2)
The question: are (X - Y) and (X - Z) independent? How can I proove it?