Some presentations of the real number field

It is proved that every two $\Sigma$-presentations of an ordered field $\mathbb R$ of reals over $\mathbb{HF(R)}$, whose universes are subsets of $\mathbb R$, are mutually $\Sigma$-isomorphic. As a consequence, for a series of functions $f\colon\mathbb R\to\mathbb R$ (e.g., $\exp$, $\sin$, $\cos$, $\ln$), it is stated that the structure $\mathbb R=\langle R,+,\times,<,0,1,f\rangle$ lacks such $\Sigma$-presentations over $\mathbb{HF(R)}$.