Additivity of independent gamma distributions with a common rate

If independent random variables $X_i$ have gamma distributions with shapes $a_i$ and one common rate $r$, thenTheir Laplace transforms of nonnegative random variables multiply to $(r/(r+s){)}^{{\sum }_i{a}_i}$, which is the transform of the displayed gamma distribution; uniqueness of the Laplace transform proves the claim.