To help with your proofs, it seems like the notation you're using contains some non-standard symbols. I'll break down the elements based on my understanding and offer a clearer path for solving it.

1. For the first statement:

   $1. \ @z\$y\$o(Fzy > Gzo) \ \text{pr} /:. \ @x\$y(Fxx > Gxy)$

   • The symbols @, $, and o likely indicate logical operations or some form of quantification.
   • The symbol pr might denote that you need to prove something or that a particular conclusion follows.
   • Fzy > Gzo seems to imply some kind of relation between terms, possibly "Fzy implies Gzo" or some inequality involving F and G.

   Possible interpretation: You're asked to prove the implication @x\$y(Fxx > Gxy) from the assumption @z\$y\$o(Fzy > Gzo).

2. For the second statement:

   2. \ @x[$y($zFzz \& y/y) > @y(Py \& x/y)] \ \text{pr}

   • The & typically indicates a conjunction (AND).
   • y/y might be shorthand for some identity or equality involving y.
   • The @ symbols and the nested quantifiers likely mean you're dealing with some form of formal logic, possibly modal logic or higher-order logic.

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Next steps:

I would suggest the following steps to proceed:

1. Define the operations clearly. We need to identify what symbols like @, $, and o represent. If they refer to quantifiers or logical operations, we can adjust accordingly.
2. Rephrase the problem in terms of standard logical syntax so we can apply formal proof strategies like deduction, contradiction, or construction of counterexamples.

Can you clarify the notation or share more details about the specific symbols you're using? Or, would you like me to proceed based on common logical operations?

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Questions:

1. What do the symbols @, $, and o represent in your notation?
2. What logical system are you working in (e.g., predicate logic, modal logic)?
3. Are there specific axioms or theorems you would like to use for these proofs?
4. Could you explain what the notation y/y means in your context?
5. Do you want me to simplify and solve the problem assuming common logical rules?

Tip:

In formal logic, always clarify the interpretation of non-standard symbols and operations before starting a proof to avoid confusion.