  We present a first principle scheme to compute the rigidity, i. e. the
shear-modulus of structural glasses at finite temperatures using the cloned
liquid theory, which combines the replica theory and the liquid theory. With
the aid of the replica method which enables disentanglement of thermal
fluctuations in liquids into intra-state and inter-state fluctuations, we
extract the rigidity of metastable amorphous solid states in the supercooled
liquid and glass phases. The result can be understood intuitively without
replicas. As a test case, we apply the scheme to the supercooled and glassy
state of a binary mixture of soft-spheres. The result compares well with the
shear-modulus obtained by a previous molecular dynamic simulation. The rigidity
of metastable states is significantly reduced with respect to the instantaneous
rigidity, namely the Born term, due to non-affine responses caused by
displacements of particles inside cages at all temperatures down to T=0. It
becomes nearly independent of temperature below the Kauzmann temperature T_K.
At higher temperatures in the supercooled liquid state, the non-affine
correction to the rigidity becomes stronger suggesting melting of the
metastable solid state. Inter-state part of the static response implies jerky,
intermittent stress-strain curves with static analogue of yielding at
mesoscopic scales.
