Well-Ordering formulation correct?

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SUMMARY

The discussion centers on the validity of a proposed theorem regarding the generalized lexicographical order on arbitrary Cartesian products. The user seeks verification of their proof, which is presented in two images linked within the post. The theorem's formulation is questioned for its correctness, and the user expresses difficulty in finding similar proofs in existing literature on set theory.

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  • Understanding of set theory concepts
  • Familiarity with Cartesian products
  • Knowledge of lexicographical ordering
  • Ability to analyze mathematical proofs
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  • Research generalized lexicographical order in set theory
  • Study existing proofs related to Cartesian products
  • Explore advanced topics in mathematical proof techniques
  • Review textbooks on set theory for similar formulations
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Mathematicians, students of set theory, and anyone interested in the foundations of mathematical proofs and ordering principles.

andytoh
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I looked for the most generalized lexicographical (dictionary) order on an arbitrary cartesian product in textbooks but I could not find it. So I posed my own theorem (in the form of a proof question). Is the formulation correct?

http://img409.imageshack.us/img409/5022/questionkk4.jpg
 
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Here's my proof. Could someone versed in set theory check if my proof is correct? I cannot find the proof in any textbook:

http://img143.imageshack.us/img143/5948/solutionbh8.jpg
 
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