Why Do Field Lines NEVER Cross?

An electric field \(\class{purple}{\boldsymbol{E}}\) indicates at every location in space the electric force that would act on a stationary charge and in which direction this force will accelerate the charge. The electric field can be visualized using field lines. If a stationary charge is placed in the field, it will be accelerated along the field line.

The question is: Why can two field lines never cross, as it is shown in Illustration 1? This question is similar to the question why there cannot be two different temperatures at one point in space.

As soon as the charge reaches the intersection of two field lines, it would no longer be clear in which of the two directions it will continue to move. Along this field line or along that field line? This contradicts the superposition principle, according to which two electric field vectors at one location would simply add up and thus result in a single, unique field vector. Thus, there would be no crossing point of the field lines, and the charge would move in the direction of the resulting force vector. Also mathematically, the crossing point would be inconsistent with the definition of a vector field, since one point in space is always assigned one unique vector.

We have only considered the example of electric field lines. The same argumentation is valid also for the magnetic field lines!