Understanding Euler's rotation theorem 
According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.  - wikipedia

so can we say that the new axis of rotation is just the resultant of the previous too ? 
Can you kindly explain this to me ?
And show ( with some examples ) how can we mathematically predict this new axis of rotation ?
 A: Let us say you have a sequence of rotations about two axes, $\hat{x}$ and $\hat{y}$ by the angles $\varphi$ and $\theta$. You use rotation matrices to find the final orientation
$$ \mathtt{E} = {\rm Rot}(\hat{x},\varphi)\,{\rm Rot}(\hat{y},\theta) $$
Now lets add some motion, and give the angles some speed
$$ \begin{align} 
  \dot{\varphi} & = \frac{{\rm d}\varphi}{{\rm d}t} \\
  \dot{\theta} & = \frac{{\rm d}\theta}{{\rm d}t}
\end{align} $$
The rotational velocity of this body is thus defined as
$$ \vec{\omega} = \hat{x} \dot{\varphi} + {\rm Rot}(\hat{x},\varphi) \hat{y} \dot{\theta} $$
This is because $\hat{y}$ rotates about $\hat{x}$.
This motion is decomposed onto a rotation axis $\hat{z}$ and a rotation magnitude $\omega$
$$  \omega = \| \hat{x} \dot{\varphi} + {\rm Rot}(\hat{x},\varphi) \hat{y} \dot{\theta} \| \\ \hat{z} = \frac{\vec{\omega}}{\omega} $$
Appendix I
To derive this you use the fact that the time derivative of a rotation matrix $\mathtt{E}$ is $$\dot{\mathtt{E}} = \vec{\omega} \times \mathtt{E}$$
Use the product rule to evaluate the left hand side (with $\mathtt{E} = \mathtt{R}_x \mathtt{R}_y$)  as $$\begin{align}\dot{\mathtt{E}} &= \left( \frac{{\rm d}}{{\rm d}t} \mathtt{R}_x \right) \mathtt{R}_y + \mathtt{R}_x \left(\frac{{\rm d}}{{\rm d}t} \mathtt{R}_y \right)\\ &= ((\hat{x} \dot{\varphi}) \times \mathtt{R}_x ) \mathtt{R}_y +\mathtt{R}_x  ( (\hat{y} \dot{\theta})\times \mathtt{R}_y) \\ &= (\hat{x} \dot{\varphi}) \times (\mathtt{R}_x  \mathtt{R}_y) + (\mathtt{R}_x \hat{y} \dot{\theta})\times (\mathtt{R}_x \mathtt{R}_y) \\ &= (\hat{x} \dot{\varphi} + \mathtt{R}_x \hat{y} \dot{\theta}) \times (\mathtt{R}_x \mathtt{R}_y) \\ &= \vec{\omega} \times \mathtt{E} \,.\end{align}$$
