Proof: By the definition of $\psi(x)$ (Equation (33)), we have
$$\sum_{m \leq x} \psi(x / m)=\sum_{m \leq x} \sum_{k \leq x / m} \Lambda(k)=\sum_{m \leq x} \sum_{k m \leqslant x} \Lambda(k),$$

By making the integer variable substitution $k m=d, k=k$, the above equation becomes
$$\sum_{m \leqslant x} \psi(x / m)=\sum_{d \leqslant x} \sum_{k \mid d} \Lambda(k)=\sum_{d \leqslant x} \ln d,$$

The last step uses Equation (38). This thus proves Equation (39).