Inverse Integral Operator is Linear if Unique

Theorem
Let $T$ be an integral operator.

Let $f$ be an integrable real function on a domain appropriate to $T$.

Let $F = T \left({f}\right)$ and $G = T \left({g}\right)$.

Let $T$ have a unique inverse $T^{-1}$.

Then $T^{-1}$ is a linear operator:
 * $\forall p, q \in \R: T^{-1} \left({p F + q G}\right) = p T^{-1} \left({F}\right) + q T^{-1} \left({G}\right)$

Proof
Let:
 * $x_1 = T^{-1} \left({F}\right)$
 * $x_2 = T^{-1} \left({G}\right)$

Thus:
 * $F = T \left({x_1}\right)$
 * $G = T \left({x_2}\right)$

Then for all $p, q \in \R$:

and so $x = p F + q G$ is a solution to the equation:
 * $T \left({x}\right) = p F + q G$

But this equation has only one solution:
 * $x = T^{-1} \left({p F + q G}\right)$

Thus $p F + q G$ must coincide with the above:
 * $p T^{-1} \left({F}\right) + q T^{-1} \left({G}\right) = T^{-1} \left({p F + q G}\right)$

which proves that $T^{-1}$ is a linear operator.