  We report CO(5$\rightarrow$4) and CO(6$\rightarrow$5) line observations in
the dusty starbursting galaxy CRLE ($z = 5.667$) and the main-sequence (MS)
galaxy HZ10 ($z = 5.654$) with the Northern Extended Millimeter Array (NOEMA).
CRLE is the most luminous $z>5$ starburst in the COSMOS field and HZ10 is the
most gas-rich "normal" galaxy currently known at $z>5$. We find line
luminosities for CO(5$\rightarrow$4) and CO(6$\rightarrow$5) of (4.9 $\pm$ 0.5)
and (3.8 $\pm$ 0.4) $\times$ 10$^{10}$ K km s$^{-1}$ pc$^{2}$ for CRLE and
upper limits of $< 0.76$ and $< 0.60$ $\times$ 10$^{10}$ K km s$^{-1}$ pc$^{2}$
for HZ10, respectively. The CO excitation of CRLE appears comparable to other
$z>5$ dusty star-forming galaxies (DSFGs). For HZ10, these line luminosity
limits provide the first significant constraints of this kind for a MS galaxy
at $z > 5$. We find the upper limit of $L'_{5\rightarrow4}/L'_{2\rightarrow1}$
in HZ10 could be similar to the average value for MS galaxies around $z\approx
1.5$, suggesting that MS galaxies with comparable gas excitation may already
have existed one billion years after the Big Bang. For CRLE we determine the
most likely values for the H$_2$ density, kinetic temperature and dust
temperature based on excitation modeling of the CO line ladder. We also derive
a total gas mass of $(7.1 \pm 1.3) \times 10^{10} M_\odot$. Our findings
provide some of the currently most detailed constraints on the gas excitation
that sets the conditions for star formation in a galaxy protocluster
environment at $z > 5$.
