ER Modeling; multiplicity basics - one to many or many to many?
I think your design could be stated more formally as:

One company has many projects. 
One project belongs to one company.
One project has one or more activities.
One activity has exactly one activity type (prototyping, design).
One activity belongs to exactly one project. 


So, if that's true, you have the following tables:

COMPANY
CompanyID
....

PROJECT
ProjectID
....

ACTIVITY_TYPE
ActivityTypeID
Name (e.g. "Prototyping", "Design")

ACTIVITY
ActivityID
ProjectID
ActivityTypeID
StartDate
EndDate
...