Trouble Understanding a Remark about Beta-Reduction in the Lambda Calculus
Application in the lambda calculus is not associative. $m_1 m_2 m_3$ means $(m_1 m_2) m_3$ and is not equal to $m_1 (m_2 m_3)$.  So the second equality you state doesn't hold. $$a(m_1 m_2 m_3) \neq ((a m_1) m_2) m_3$$  Further $$a N_1 N_2 = (a m_1)(m_2 m_3) \neq (a m_1) m_2 m_3 = ((a m_1) m_2) m_3$$