If the standard reals and hyperreals are non-isomorphic structures, proofs transferred between them establish facts about different mathematical objects, undermining the claim of genuine simplification.

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hyperreals(Nonstandard analysis): A number system developed by Robinson that extends the standard reals and satisfies the transfer principle with respect to first-order statements about the reals
isomorphic(Model theory): A structural identity between two mathematical objects where there exists a bijection preserving all relevant structure
non-isomorphic structures(as used in mathematics and logic): Two mathematical systems that have fundamentally different shapes or patterns—they can't be perfectly matched up or translated into each other, even though they might seem similar on the surface.
reals (standard reals)(as used in mathematics and logic): The regular numbers you learn in school—integers, fractions, decimals, and irrational numbers like π. They're called 'standard' to distinguish them from other number systems mathematicians have invented.

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