Result on D3306: Basic real harmonic mean
Basic real harmonic and arithmetic means are multiplicative inverses when arguments are
Formulation 0
Let $x_1, \dots, x_N \in (0, \infty)$ each be a D993: Basic real number such that
(i) $A(x_1, \ldots, x_N)$ is a D2454: Basic real arithmetic mean of $x_1, \dots, x_N$
(ii) $H \left( \frac{1}{x_1}, \ldots, \frac{1}{x_N} \right)$ is a D3306: Basic real harmonic mean of $\frac{1}{x_1}, \ldots, \frac{1}{x_N}$
Then \begin{equation} H \left( \frac{1}{x_1}, \ldots, \frac{1}{x_N} \right) A(x_1, \ldots, x_N) = 1 \end{equation}
Proofs
Proof 0
Let $x_1, \dots, x_N \in (0, \infty)$ each be a D993: Basic real number such that
(i) $A(x_1, \ldots, x_N)$ is a D2454: Basic real arithmetic mean of $x_1, \dots, x_N$
(ii) $H \left( \frac{1}{x_1}, \ldots, \frac{1}{x_N} \right)$ is a D3306: Basic real harmonic mean of $\frac{1}{x_1}, \ldots, \frac{1}{x_N}$
Results
(i) R4095: Basic real arithmetic expression for basic real arithmetic mean
(ii) R4092: Basic real arithmetic expression for basic real harmonic mean

provide the expressions \begin{equation} A(x_1, \ldots, x_N) = \frac{1}{N} \sum_{n = 1}^N x_n , \qquad H (x_1, \ldots, x_N) = \frac{N}{\sum_{n = 1}^N \frac{1}{x_n}} \end{equation} Thus \begin{equation} \begin{split} H \left( \frac{1}{x_1}, \ldots, \frac{1}{x_N} \right) = \frac{N}{\sum_{n = 1}^N \frac{1}{1 / x_n}} = \frac{N}{\sum_{n = 1}^N x_n} = \frac{1}{\frac{1}{N} \sum_{n = 1}^N x_n} = \frac{1}{A(x_1, \ldots, x_N)} \end{split} \end{equation} Multiplying both sides by $A(x_1, \ldots, x_N)$ finishes the proof. $\square$