Is this a theorem?

Given any integer $m$, there exists $N(m)$, such that for all $n>N(m)$, we have $$n = a_1+a_2+\cdots+a_m$$ where $\gcd(a_i,a_j) = 1$ with $a_i>1$ for all $i,j>1$.

For instance, it is a textbook exercise to show for small values of $m$; Any integer exceeding $6$ is a sum of two relatively prime numbers; Any integer exceeding $17$ is a sum of three relatively prime numbers and so on.