Kirchhoff voltage law indicates that :the sum of the potential diffrence across any closed loop will be equal to zero which can be written as $\sum_{i=1}^{n}V_{i}=0$
but what is the proof of this law ?
all I can think of is that due to we are dealing with an Electrostatic field so the closed line integral through any loop will equal to zero which is
$\oint_{C}^{}\overrightarrow{E}.d\overrightarrow{L}=0$
Contrary to the comment by @march above to establish "the" KVL, Kirchhoff's Voltage Law, it is not enough to say that the field be conservative, i.e., $\oint \bf E \cdot d\ell = 0$ because as stated this would exclude all magnetic phenomena, for example, such as induction. To include Faraday induction you must also assume that all two-terminal elements, $R,L,C$ ($v=Ri$, $v=Ldi/dt$ and $i=Cdv/dt$) you connect with ideal conductors are lumped so that their behavior can be represented by the respective input=output current and voltage drop between the terminals. Consequently the magnetic field, if any, must also be concentrated within such element that we call an inductor coil. You may also include magnetic couplings between elements by assuming more complicated port relationships, a transformer, but again these must be concentrated to the couplings.
Once you do so, you can write the loop integral of the $E$ field as you have but then it becomes not really a physical law as such but rather a compatibility requirement for voltage drops in any loop over elements that may form your network as part of a larger network consisting also other loops. The same consideration goes for KCL that requires that if there is a junction/node where charges may accumulate during which you may have temporarily $\sum_k i_k \ne 0$ you introduce a new lumped element to act as a current sink/source so that the now the currents do satisfy KCL.
If you really want to get extreme then you may say that neither KCL nor KVL is really physics but rather a topological theorem that allows proper assignment of a set of numbers to the branches and nodes of a graph. For details of such viewpoint and its importance see Tellegen's Theorem
