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In mathematical logic, a theory can be extended withnew constants or function names under certain conditions with assurance that the extension will introduceno contradiction. Extension by definitions is perhaps the best-known approach, but it requiresunique existence of an object with the desired property. Addition of new names can also be donesafely without uniqueness.
Suppose that a closed formula
is a theorem of a first-order theory T {\displaystyle T}. Let T 1 {\displaystyle T_{1}} be a theory obtained from T {\displaystyle T} by extending its language with new constants
and adding a new axiom
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