The 'progress' practice in PRINCE2 should be used to monitor and control the performance of the project against its plans and the project's overall direction. This includes predicting whether the benefits will exceed the costs, which is crucial for evaluating the project's ongoing viability and justifying the continuation of the project based on its forecasted value to the organization.References: (PRINCE2 7 Foundation Handbook, AXELOS)

The objectives of the 'Starting Up a Project' process include providing an interface between the corporate or program environment and the project (objective 1) and providing business justification for initiating the project (objective 4). This process ensures that the project is viable and worthwhile before further resources are committed and that there is a clear link to corporate strategies and ongoing work.References: (PRINCE2
7 Foundation Handbook, AXELOS)

In PRINCE2, the "Closing a Project" process has the objective of confirming that the project's products are acceptable and meet the established acceptance criteria. This process ensures that the project has delivered the required outputs as agreed upon, and these outputs are approved and accepted by the customer. References:
PRINCE2 Foundation Guide, "Closing a Project" section

The purpose of the 'Directing a Project' process in PRINCE2 is to enable the project board to be accountable for the project's success by making key decisions and exercising overall control while delegating the day-to-day management of the project to the project manager. This includes the authority to initiate and close the project, deliver stage boundaries, and be updated on project progress.References: = PRINCE2 7 Foundation handbook and online resources

In the PRINCE2 methodology, the 'project mandate' is the term used for initial documentation that triggers the start of a project, typically provided by corporate or program management. It can often come in the form of a request for proposal received by an external supplier, setting out the initial definition of the project and what it aims to achieve, which then leads to more detailed project definitions in the 'starting up a project' process.
References: PRINCE27 Foundation resources on project initiation documentation.