If $x_1(n) (\lt-\gt)^{DFT} x_1(k)$ And

$x_2(n) (\lt-\gt)^{DFT} x_2(k)$ Then

$a_1x_1(n) + a_2x_2(n) (\lt-\gt)^{DFT} a_1x_1(k) + a_2x_2(k)$

Proof:- By the definition

x(k) = ∑ x(n) $(W_n)^{Kn}$

here x(n) = $a_1x_1(n)$ + $a_2x_2(n)$

=> X(K) = ∑[$a_1x_1(n)$ + $a_2x_2(n)$]$(W_n)^{Kn}$

X(K) = $a_1X_1(k)$ + $a_2X_2(k)$

Hence proved.