Final answer:
The procedure of washing hands after greeting a client is known as hand hygiene, which is essential to prevent the spread of infections and maintain cleanliness.
Step-by-step explanation:
The procedure described in the question, where the petitioner (often a healthcare professional or someone in a setting that requires cleanliness) greets a client and then washes their hands, is commonly referred to as hand hygiene. This is a basic but critical practice in healthcare and other professions to prevent the spread of germs and infections. The act of washing hands after shaking hands with a client or patient is considered best practice to maintain a sanitary environment.