# Decomposition of real algebraic varieties into manifolds

I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry.

I am dealing with a (real) algebraic variety $$V$$ of (Krull) dimension $$n$$. I keep reading that $$V$$ can be decomposed into differentiable manifolds of dimensions $$0, 1, ..., n$$.

While this seems plausible I could not find this statement in the literature. Is it true? I would also be glad about a reference.