Mathematic notation of scale returns
Quoting from the question:


  
  $F(aK,aL)$ : simultaneous and proportional inputs increase
  $aF(K,L)$ : output increase
  


But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if it was increased by the same proportion as the inputs. So
\begin{align*}
F(aK,aL) &amp; &gt; aF(K,L)
\end{align*}
shows that if you increase the inputs simultaneously and proportionally, the increase in output is larger than proportional.