Real Inverse Hyperbolic Sine Function is Bijection

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Theorem

The real inverse hyperbolic sine is a bijection.


Proof

From Hyperbolic Sine is Bijection over Reals and by definition of bijection, we have that $\sinh$ admits an inverse function over $\R$.

From:

Domain of Bijection is Codomain of Inverse
Codomain of Bijection is Domain of Inverse

the domain and image of hyperbolic sine over $\R$, is $\R$.

$\blacksquare$