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