To prove that a composition is a mapping, we need to show that for any two elements a and b in the domain of the composition, the composition f(g(a)) and f(g(b)) are well-defined and unique. This means that for any input a, there is a unique output f(g(a)). We also need to show that the composition is closed under the given operation, meaning that the composition of any two elements in the domain results in an element in the codomain. Finally, we need to show that the composition satisfies the properties of a mapping, such as associativity and identity. By demonstrating these properties, we can prove that the composition is indeed a mapping. **

Uploading a music composition on YouTube can serve as evidence of authorship, as it provides a public record of the creation and ownership of the work. However, it may not be sufficient on its own to establish legal proof of authorship. It is important to also consider other forms of documentation, such as copyright registration, to fully protect the rights of the creator. Additionally, it is advisable to include clear copyright notices and information about the author in the video description or accompanying materials to further establish authorship. **

The proof for the composition of two linear functions lies in the fact that the composition of two linear functions is also a linear function. This can be shown using the properties of linearity, which state that a function is linear if it satisfies the properties of additivity and homogeneity. When two linear functions are composed, the resulting function still satisfies these properties, and therefore is also a linear function. This can be proven algebraically by applying the definition of linearity to the composition of the two functions. **

Escape-proof prisons are designed to prevent inmates from breaking out, but that doesn't necessarily mean they are burglary-proof. While the security measures in place may be effective at keeping prisoners in, they may not be as effective at keeping burglars out. Burglary-proofing a facility requires different security measures, such as alarms, surveillance cameras, and secure entry points, which may not be the primary focus of an escape-proof prison. Therefore, just because a prison is escape-proof doesn't guarantee that it is also burglary-proof. **

Thomas Aquinas' contingency proof is based on the idea that everything in the world is contingent, meaning it relies on something else for its existence. Therefore, there must be a necessary being, or God, that is the ultimate cause of all contingent beings. Aquinas' cosmological proof, on the other hand, argues that everything in the world is in motion or change, and there must be a first mover, or God, that initiated this motion. Both proofs aim to demonstrate the existence of God through logical reasoning and observation of the natural world. **

No, Gödel's proof is not a proof of God's existence. Gödel's ontological proof is a mathematical argument that attempts to show that the existence of God is a necessary consequence of a certain set of axioms and definitions. However, the proof has been widely debated and criticized, and it is not widely accepted as a definitive proof of God's existence. The question of God's existence is a complex and deeply philosophical one that cannot be settled by a single mathematical proof. **

The profession of bookseller is facing challenges in the digital age, with the rise of e-books and online retailers. However, there is still a demand for physical books and the experience of browsing in a bookstore. To remain relevant, booksellers may need to adapt by offering a unique and personalized experience, hosting events, and diversifying their offerings. While the profession may not be crisis-proof, there is still potential for success by embracing change and finding ways to connect with readers. **

Professions in the healthcare and technology sectors are considered to be future-proof and crisis-proof. Healthcare professionals, such as doctors, nurses, and healthcare administrators, will always be in demand as the population continues to grow and age. Similarly, technology professionals, such as software developers, data analysts, and cybersecurity experts, are essential in our increasingly digital world. These professions are likely to remain in high demand regardless of economic fluctuations or crises. **