Proof of superposition for linear circuit

Linear circuit must meet the superposition theorem,while circuit that meets the superposition theorem must be linear circuit,but circular definition does not help in identifying linear circuits or applying superposition theorem.This article points out that linearity is a description for the input and output signal relation of linear system,which is the sum of homogeneity and additivity.Linear circuit element is any circuit element whose circuit/voltage relation meet the requirement of linear system input/output relation,which can be further divided into ordinary linear circuit element whose circuit and voltage relation of itself satisfy linearity,and controlled linear circuit element whose controlled signal and control signal relation satisfy linearity.Linear circuit is composed by linear element and independent source wherein independent source is seen as input(excitation) of linear circuit while any voltage and circuit are seen as output(response).Finally a proof is given based on the concept of linearity and Kirchhoff's Law.