Proof of the superposition theorem in linear circuit analysis
A linear system in electronics means a system that can be expressed as a linear differential equation. Hence

$$\dot{x} = \mathbf{A} x + \mathbf{B}u,$$

where \$x\$ is the state vector, \$u\$ is the input vector, and \$\mathbf{A}\$ and \$\mathbf{B}\$ are matrices. Let \$x_1\$ be the state from \$u_1\$ and let \$x_2\$ be the state from \$u_2\$. Thus

$$\frac{\mathrm{d}}{\mathrm{d}t} ( x_1 + x_2) = \dot{x}_1 + \dot{x}_2 = \mathbf{A} x_1 + \mathbf{B}u_1 + \mathbf{A} x_2 + \mathbf{B}u_2.$$

Therefore
$$\frac{\mathrm{d}}{\mathrm{d}t} ( x_1 + x_2) = \mathbf{A} (x_1 + x_2) + \mathbf{B}(u_1 + u_2).$$

This means that applying both inputs together is equivalent to adding the states for each input applied individually. For superposition of more than two inputs apply induction.