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We report CO(5 → 4) and CO(6 → 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. CRLE is the most luminousz> 5 starburst in the COSMOS field and HZ10 is the most gas-rich “normal” galaxy currently known atz> 5. We find line luminosities for CO(5 → 4) and CO(6 → 5) of (4.9 ± 0.5) and (3.8 ± 0.4) × 10¹⁰K km s⁻¹pc²for CRLE and upper limits of < 0.76 and < 0.60 × 10¹⁰K km s⁻¹pc²for HZ10, respectively. The CO excitation of CRLE appears comparable to otherz> 5 dusty star-forming galaxies. For HZ10, these line luminosity limits provide the first significant constraints of this kind for an MS galaxy atz> 5. We find the upper limit of$L_{5\to 4}^{\prime }/L_{2\to 1}^{\prime }$in HZ10 could be similar to the average value for MS galaxies aroundz≈ 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 H₂density, kinetic temperature, and dust temperature based on excitation modeling of the CO line ladder. We also derive a total gas mass of (7.1 ± 1.3) × 10¹⁰M_⊙. 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 atz> 5.