Primal-dual inner point algorithms are known to be efficient in solving non-linear constrained optimization problems. Modern implementations are capable of solving optimization problems with a huge number of non-linear constraints. To do this efficiently it is crucial, that necessary optimality conditions are formulated such that they can be easily implemented into a computer program. Favourable is a formulation as a system of equations that can be linearized. The Karush-Kuhn-Tucker conditions represent such a set. This work gives a rigours proof for the equivalence of the necessary conditions of the reformulations of a non-linear constrained optimization problem as they are used in inner point methods.