I've been struggling with understanding of the proofs showing that implementation statements are extensional.Basically I am referring to material described in Abstraction Classes in Software Design paper.As an example, let us take the following proof for class diagram Figure 13Proposition 11. The class diagram in Figure 13 is extensional.Proof: Let us show that Figure 13 is preserved under expansion. Let M designate a structure that satisfies Figure 13. Any such structure shall include,   among others, the following entities and relations:Class entities, such as Creator and ConcreteCreator, as well as   Method entities, such as Creator::FactoryMethod and   ConcreteCreator::FactoryMethod. The binary relation Member, which associates every method with one   class, such as the pair (Creator::FactoryMethod, Creator)The binary relation Inherit, which includes the pairs   (ConcreteCreator, Creator) and (ConcreteProduct, Product)Additional information in Figure 13 is represented informally. For  example, the  relation Invoke(AnOperation,FactoryMethod) is  represented by the note product=FactoryMethod() at the top right  corner of the diagram.Note that every element in the diagram  explicitly represents a specific entity (e.g., class Creator) or a  specific relation (e.g., Inherit(ConcreteCreator, Creator)). Thus,  expanding M with any number of Class entities cannot violate Figure  13.Let us show that Figure 13 is preserved under reduction. There are two types of entities that can be removed from M:Entities that are explicitly mentioned in Figure 13.Entities that are not explicitly mentioned in Figure 13.By the Intension criterion, we  only consider reductions removing entities  that are not explicitly  mentioned in Figure 13 from the universe of M. Obviously, any such  reduction will continue to satisfy Figure 13.My specific question is about preserving Figure 13 under reduction. How can we remove entities that are not explicitly mentioned in Figure 13? What are these entities that are not explicitly mentioned in Figure 13? Just don't get how can we remove something that is not present on the diagram.
Proofs for Implementation Statements
lo.logic;se.software engineering