= Solution
For total relativistic <energy> $E=\gamma mc^2$, the speed is $v=c\sqrt{1-(mc^2/E)^2}$. Both beam energies are enormously greater than the rest <energy>, so expanding the square root avoids obscuring the small difference:
$$
\Delta v\simeq\frac c2(mc^2)^2\left(\frac1{E_1^2}-\frac1{E_2^2}\right).
$$
With all energies in GeV, $mc^2=0.9383$, $E_1=4000$ and $E_2=6500$, giving
$$
\boxed{\Delta v\simeq5.12\,\mathrm{m\,s^{-1}}.}
$$
The deficits below the speed of light are about $8.25\,\mathrm{m\,s^{-1}}$ and $3.12\,\mathrm{m\,s^{-1}}$. The large <energy> increase therefore produces only a small speed increase; the <energy> mainly changes the <Lorentz factor>. These numbers use the quoted beam energies as total energies. Interpreting them as kinetic energies requires adding the rest <energy>, with only a tiny correction at these scales.
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