Unilateral Laplace transform is a linear operator

Theorem

If $\alpha, \beta \in \mathbb{C}$ and $f, g \colon \mathbb{T} \rightarrow \mathbb{C}$ have unilateral Laplace tarnsforms, then $$\mathscr{L}_{\mathbb{T}}\{\alpha f + \beta g\} = \alpha \mathscr{L}_{\mathbb{T}}\{f\} + \beta \mathscr{L}_{\mathbb{T}}\{g\}.$$