Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

The suction and delivery of air takes place at constant pressure
There is no pressure drop in the inter-cooler
All of the listed here
The compression in both the cylinders is polytropic

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Compressors, Gas Turbines and Jet Engines
For minimum work in multistage compression, assuming same index of compression in all stages

Work done in all stages should be equal
Work done in subsequent stages should increase
Work done in first stage should be more
Work done in subsequent stages should decrease

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