conventions - Why can we assume a current direction when using kirchhoff's circuit law?

Reference direction: When analyzing electrical circuits, the actual direction of current through a specific circuit element is usually unknown. Consequently, each circuit element is assigned a current variable with an arbitrarily chosen reference direction. When the circuit is solved, the circuit element currents may have positive or negative values. A negative value means that the actual direction of current through that circuit element is opposite that of the chosen reference direction.

Why can we assume a direction and get the correct value and sign? Is there a simple proof of this fact?

Answer

Is there a simple proof of this fact?

Sure; when you insert an ammeter into a circuit branch, there are two choices of polarity which amount to choosing a reference direction.

If two identical ammeters, connected in series with opposite polarity, are inserted into a circuit branch, they will measure the same current but give the opposite sign since each has a different reference direction.

However, they both give the same information. For one ammeter, current enters the positive lead and this ammeter gives a positive reading. For the other ammeter, current exits the positive lead and this ammeter gives a negative reading.

In either case, the ammeter reading gives you the correct direction of the current.