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:
 * $\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \, \map T f + \beta \, \map T g$

Proof
Let $T$ be expressed in its full form as an integral fransform:
 * $\map T f := \displaystyle \int_a^b \map f x \, \map K {p, x} \rd x$

for some integrable function $\map K {p, x}$.

Then: