  Let $\vec{X}=(X_0, X_1)$ be a compatible couple of Banach spaces, $ 1\le p
\le \infty$ and let $ \varphi$ be positive quasi-concave function. Denote by
$\overline{X}_{\varphi,p}=(X_0,X_1)_{\varphi,p}$ the real interpolation spaces
defined by S. Janson (1981). We give necessary and sufficient conditions on $
\varphi_{0}$, $\varphi_{1}$ and $\varphi$ for the validity of \begin{equation*}
  \left(\overline{X}_{\varphi_{0},1},\overline{X}_{\varphi_{1},1}
  \right) _{\varphi,p}=
\left(\overline{X}_{\varphi_{0},\infty},\overline{X}_{\varphi_{1},\infty}\right)_{\varphi,p}
\end{equation*} for all $ 1\le p\le \infty$, and all Banach couples
$\overline{X}. $
