= Solution
At any chosen point take <normal coordinates> for $g$, so $\Gamma=0$ there. Substitute $\widetilde\Gamma=\Gamma+T$ into the stated <Riemann curvature tensor> formula and subtract the original curvature. At that point the mixed products with $\Gamma$ vanish and ordinary derivatives of $T$ equal its covariant derivatives. The result is
$$
\widetilde R^a{}_{bcd}-R^a{}_{bcd}=T^a{}_{bd;c}-T^a{}_{bc;d}-T^a{}_{ed}T^e{}_{cb}+T^a{}_{ec}T^e{}_{db}.
$$
In unit-weight antisymmetrization notation this is \b[$\boxed{2T^a{}_{b[d;c]}-2T^a{}_{e[d}T^e{}_{c]b}}$]. Every term is a tensor, so the identity established in normal coordinates holds in every coordinate system.
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