= Solution
If the discounted <zero-coupon bond> price is a <martingale>, its <expectation> is $Z_{0,T}=e^{-A(0,T)}$. The <Gaussian moment-generating function> applied to the representation above gives $\mathbb E Z_{s,T}=e^{-A(s,T)+v(s,T)/2}$, so
$$
A(s,T)-A(0,T)=\frac12v(s,T).
$$
Differentiate in maturity $T\geq s$. The two mean derivatives are $\mu_{s,T}$ and $\mu_{0,T}$. Symmetry of the <covariance> in its last two arguments makes the two boundary terms in the derivative of the double integral equal. Hence
$$
\frac12\partial_Tv(s,T)=\int_0^T c(s\wedge u,u,T)\,du.
$$
We obtain the <Gaussian forward-rate covariance drift restriction>
$$
\boxed{\mu_{s,T}=\mu_{0,T}+\int_0^T c(s\wedge u,u,T)\,du,}
$$
which proves that the <martingale> condition implies the mean restriction. Only maturity differentiation is used; a time derivative of the <covariance> is unnecessary.
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