  In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 :
{\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian
symmetric space $M$ of compact type in the case where $L_0$ is not necessarily
congruent to $L_1$. In particular, we have a generalization of the
Arnold-Givental inequality in the case where $M$ is irreducible. As its
application, we prove that the totally geodesic Lagrangian sphere in the
complex hyperquadric is globally volume minimizing under Hamiltonian
deformations.
