Can the organisation servicing my equipment also carry out the thorough examination?

Yes. But the competent person undertaking the thorough examination should not be the same person who services / maintains the equipment, as they would be responsible for assessing their own work. A competent person should be sufficiently independent and impartial in to make objective decisions. Although a competent person that undertakes thorough examinations is usually employed by a different company, they may be employed by the equipment user, provided they have sufficient independence to act impartially.