Recall the definition of a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$ and the theorem characterizing such a diffeomorphism among applications of class $\mathcal { C } ^ { 1 }$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R } ^ { 2 }$.