On the C*-valued triangle equality and inequality in Hilbert C*-modules

We prove that for two elements x, y in a Hilbert A-module V the triangle equality |x+y|=|x|+|y| holds if and only if the inner product of x and y is equal to |x||y|. In addition, we establish some kind of a C*-valued triangle inequality in a Hilbert C*-module over a unital C*-algebra.