NOVELTY - Separating a mixture of gases or liquids comprises: sending mixture of gases or liquids to pre-membrane unit (113) to separate them into first permeate stream passing through membrane and first retentate stream that does not pass through membrane; sending first retentate stream to primary membrane unit (121) (comprising glassy polymer membrane) to give second permeate stream and second retentate stream; and then sending second permeate stream to secondary membrane unit (135) (comprising glassy polymer membrane) to give third permeate stream and third retentate stream.    USE - The process is useful for separating mixtures of gases or liquids.    ADVANTAGE - The process: exhibits high selectivity and high permeance by incorporating high performance polybenzoxazole polymer and cross-linked polybenzoxazole polymer membranes, which are having easy processability, high selectivity, high permeability, high thermal stability and high resistance to solvent swelling, plasticization and hydrocarbon contaminants; reduces inter-stage compression cost; and exhibits increased product recovery with high product purity.    DETAILED DESCRIPTION - Separating a mixture of gases or liquids, comprises: sending the mixture of gases or liquids to a pre-membrane unit (113) to separate them into a first permeate stream passing through a membrane within the pre-membrane unit and a first retentate stream that does not pass through the membrane within the pre-membrane unit, where the membrane within the pre-membrane unit has selectivity at least greater than 20; sending the first retentate stream to a primary membrane unit (121) to separate the first retentate stream into a second permeate stream and a second retentate stream, where the primary membrane unit comprises at least one glassy polymer membrane; and then sending the second permeate stream to a secondary membrane unit (135) to separate it into a third permeate stream and a third retentate stream, where the secondary membrane unit comprises at least one glassy polymer membrane. An INDEPENDENT CLAIM is also included for a system for separation of mixtures of gases or liquids comprising a pre-membrane unit, a primary membrane unit and a secondary membrane unit, where the pre-membrane comprises a membrane with selectivity at least higher than 20, the primary membrane comprises at least one glassy polymer, and a liquid or gas mixture is separated into a retentate stream and a permeate stream by each of the pre-membrane unit, the primary membrane unit and the secondary membrane unit in sequence.    DESCRIPTION OF DRAWING(S) - The figure shows a multi-stage membrane system with the retentate as the product.    Feed (100)    Pre-membrane unit (113)    Primary membrane unit (121)    Compressor (130)    Secondary membrane unit (135)