Can someone help me to prove this statement? Or provide argumentation why it holds.
$$\frac{\partial x^a}{\partial x^{b'}}\frac{\partial x^{b'}}{\partial x^c}=\delta^{a}_{b}$$
It uses the Einstein summation convention - if an index is repeated, we assume the quantity is summed over. The superscripts denote indices, not exponents. The $x'$ are a different coordinate system from the $x$.
If $x^a = x^a(x')$ then using the chain-rule
$$ \delta_b^a = \frac{\partial x^a}{\partial x^b} = \frac{\partial x^a}{\partial x'^c}\frac{\partial x'^c}{\partial x^b} $$