_ _ _ _ _ _ _ _ _ _ _    TC13: 9*∀z (h(z) ∧ s(z)) → ∃z m(z)h(y) → m(y)∀x(h(x) → m(x))h(y) ∧ s(y) → m(y)¬h(y) ∨ m(y)¬(h(y) ∧ s(y)) ∨ m(y)

To analyze the given logical statement step by step, let's break it down.

1. Break Down the Problem

The expression you provided contains several logical operators and quantifiers. We need to analyze it carefully to understand its components and implications:

1. $TC13: 9 * \forall z (h(z) \land s(z)) \rightarrow \exists z m(z) h(y) \rightarrow m(y) \forall x (h(x) \rightarrow m(x)) h(y) \land s(y) \rightarrow m(y) \neg h(y) \lor m(y) \neg (h(y) \land s(y)) \lor m(y)$

2. Relevant Concepts

• Quantifiers:
  • $\forall$: For all.
  • $\exists$: There exists.
• Logical Operators:
  • $\land$: And
  • $\lor$: Or
  • $\neg$: Not
  • $\rightarrow$: Implies

3. Analysis and Detail

The statement contains multiple parts connected logically. Let's analyze each implication:

1. Initial Part: $9 * \forall z (h(z) \land s(z))$

   • This states that for every $z$, both $h(z)$ and $s(z)$ are true.

2. Implication: $\rightarrow \exists z m(z)$

   • This leads us to deduce that if the above is true, there exists some $z$ such that $m(z)$ is true.

3. Continuing with $h(y) \rightarrow m(y)$:

   • If $h(y)$ holds for some $y$, then $m(y)$ must also hold.

4. The statement $\forall x (h(x) \rightarrow m(x))$:

   • This implies that for all $x$, if $h(x)$ is true, then $m(x)$ is true.

5. Last parts involve multiple logical clauses:

   • $h(y) \land s(y) \rightarrow m(y)$:
     • Suggests that if both $h(y)$ and $s(y)$ are true, then $m(y)$ must also be true.
   • $\neg h(y) \lor m(y) \neg (h(y) \land s(y)) \lor m(y)$:
     • This is a statement involving the negation and combines the truth of $m(y)$ with the falsehood of $h(y)$ and $s(y)$.

4. Verify and Summarize

The overall flow of this logical structure suggests a relationship between the properties of elements $y$ and $z$ under $h$ and $s$ leading to conclusions about $m$. This kind of mapping can often be used in proofs or interpretations in mathematical logic, especially in predicate logic.

Final Answer

The logical expression represents complex implications among the predicates $h$, $s$, and $m$, mainly focusing on universal and existential quantifiers that dictate the truth of $m(y)$ based on the conditions provided by $h(y)$ and $s(y)$. The final logical conclusion can be derived based on the truth of each component, ensuring consistency with the implications established.