Equivalent criterion of Haar and Franklin systems in symmetric spaces

In the present article it is proved that if the Haar and Franklin systems are equivalent in a separable symmetric space $E$, the following condition holds:
\begin{equation} 0<\alpha_E\leqslant\beta_E<1, \end{equation}
where $\alpha_E$ and $\beta_E$ are the Boyd indices of the space $E$. It is already known that if condition (1) is fulfilled, it follows that the Haar and Franklin systems are equivalent in the space $E$. Thereby, this estabishes that condition (1) is necessary and sufficient for the equivalence of the Haar and Franklin systems in $E$.
In proving the assertion a number of interesting constructions involving Haar and Franklin polynomials are presented and upper and lower bounds on the Franklin functions applied.