| 1. | This is used to show existence theorems for holomorphic functions.
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| 2. | Dixon s proof used the Existence Theorem of Partha Dasgupta and Eric Maskin.
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| 3. | The existence of parallel transport follows from standard existence theorems for ordinary differential equations.
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| 4. | The Model Existence Theorem and its proof can be formalized in the framework of Peano arithmetic.
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| 5. | The crucial Takagi existence theorem was known by 1920 and all the main results by about 1930.
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| 6. | When " F " is single valued, this is the classic Peano existence theorem.
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| 7. | The actual isomorphism used and the existence theorem is described in the theory of the norm residue symbol.
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| 8. | This theorem can be considered as an existence theorem : any stationary process has this seemingly special representation.
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| 9. | However, this existence theorem is purely theoretical, as such a base has never been explicitly described.
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| 10. | An even more general result is the Carath�odory existence theorem, which proves existence for some discontinuous functions ?.
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