  The present paper provides a proof of $i^*( \mathbb{C}P+\mathbb{C}(I-P);
\mathbb{C}Q+\mathbb{C}(I-Q) )=-\chi_{orb}(P,Q)$ for two projections $P,Q$
without any extra assumptions. An analytic approach is adopted to the proof,
based on a subordination result for the liberation process of symmetries
associated with $P,Q$.
