Integral Operator is Linear

Theorem
Let $T$ be an integral operator.

Let $f$ and $g$ be integrable real functions on a domain appropriate to $T$.

Then $T$ is a linear operator:
 * $\forall \alpha, \beta \in \R: T \left({\alpha f + \beta g}\right) = \alpha T \left({f}\right) + \beta T \left({g}\right)$

Proof
Let $T$ be expressed in its full form as an integral fransform:
 * $T \left({f}\right) := \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$

for some integrable function $K \left({p, x}\right)$.

Then: