Flow path: LP compressor → intercooler → HP compressor → combustor → turbine. On the T-s diagram the "intercooling notch" between the two compression stages shows the rejected heat and the resulting work saving.
$$w_{c,IC}=c_p\big[(T_{2}-T_1)+(T_{2}'-T_{i})\big],\qquad r_{stage}=\sqrt{r_p}$$
Compressor work with intercooling, w_{c,IC}. T₂ is the stage-1 exit temperature, T_i the temperature after intercooling, T₂' the stage-2 exit temperature. The work saving is greatest when each stage takes the square root of the overall pressure ratio r_p and the intercooler returns the air to near the inlet temperature.
$$T_2=T_1\,r_p^{(\gamma-1)/2\gamma},\qquad T_i=T_2-\varepsilon\,(T_2-T_1)$$
Stage-1 exit temperature T₂ and the temperature after intercooling T_i. Each stage's isentropic temperature factor uses the square-root exponent; the higher the effectiveness ε, the closer T_i is to the inlet temperature T₁.
$$w_t=c_p\,T_3\Big(1-r_p^{-(\gamma-1)/\gamma}\Big),\qquad w_{net}=w_t-w_{c,IC}$$
Turbine work w_t and net work w_net. T₃ is the turbine inlet temperature and c_p the specific heat at constant pressure. Every bit of compressor work cut adds directly to the net work.