NOVELTY - A chemically cross-linked aromatic polyimide polymer (A1) is new.    USE - As chemically cross-linked aromatic polyimide polymer useful for making chemically cross-linked aromatic polyimide membrane for separating at least one gas from a mixture of gases e.g. propylene and propane (claimed); in the purification, separation or adsorption of a particular species in the liquid or gas phase; for the desalination of water by reverse osmosis or for the separation of proteins or other thermally unstable compounds, e.g. in the pharmaceutical and biotechnology industries; in fermenters and bioreactors to transport gases into the reaction vessel and transfer cell culture medium out of the vessel; for the removal of microorganisms from air or water streams, water purification, ethanol production in a continuous fermentation/membrane pervaporation system, and in detection or removal of trace compounds or metal salts in air or water streams.    ADVANTAGE - The high hydrocarbon resistant chemically cross-linked aromatic polyimide membrane comprising the chemically cross-linked aromatic polyimide polymer (A1) exhibits high permeability and selectivity in separation of mixtures of gases and liquids. The high hydrocarbon resistant chemically cross-linked aromatic polyimide membrane can be fabricated into any convenient geometry such as flat sheet (or spiral wound), tube, or hollow fiber.    DETAILED DESCRIPTION - A chemically cross-linked aromatic polyimide polymer (A1) having formula (I) is new.    R=H, -COCH3 and/or a compound of formula (Ia);    X1-X4= compound of formula (Ib)-(If), phenyl-1,2, 4, 5-tetra-yl, naphthalene or biphenyl;    R1a=C(=O), S(=O)2, C(CF3)2, C(CH3)2, O or S;    Y2=phenyl-1,2, 4, 5-tetra-yl, biphenyl-3,4,3',4'-tetra-yl or compound of formula (Ig);    R'=a group of formula -C(CF3)2-, -C(CH3)2-, -S(=O)2-, -O- or -S-;    Y1=2,4,6-trimethyl-phenyl-1,3-diyl, biphenyl compound of formula (Ih)-(Ik), 2,3,5,6-tetramethyl-phenyl-1,4-diyl, 3,5,3',5'-tetramethyl-biphenyl-4,4'-diyl, phenyl-1,4-diyl, biphenyl-4,4'-diyl, benzoic acid (attachment is at 3,5 positions), 4-methyl-phenyl-1,3-diyl, or compound of formula (Im);    Y3=Y1, or compounds of formulae (IIa)-(IIc);    R''=H or -COCH3;    n, m, n' and m'=2-500;or    n/m or n'/m'=1:100-100:1.    INDEPENDENT CLAIMS are included for the following:    (1) a chemically cross-linked aromatic polyimide membrane comprising the chemically cross-linked aromatic polyimide polymer (A1);    (2) an aromatic polyimide blend comprising a first aromatic polyimide containing carboxylic acid groups having several repeating units of formula (II), and a second aromatic polyimide containing hydroxyl groups having several repeating units of formula (III);    (3) an aromatic polyimide blend membrane comprising the aromatic polyimide blend;    (4) preparation (P1) of a chemically cross-linked polyimide polymer membrane involving making an aromatic polyimide blend membrane comprising a first aromatic polyimide polymer and a second aromatic polyimide polymer and then chemical cross-linking the aromatic polyimide blend membrane by heating the aromatic polyimide blend membrane at 100-300 degrees C under an inert atmosphere; and    (5) separating (P2) at least one gas from a mixture of gases involving a) providing a chemically cross-linked aromatic polyimide membrane comprising (I), b) contacting the mixture of gases to one side of the chemically cross-linked aromatic polyimide membrane to cause at least one gas to permeate membrane, and c) removing from an opposite side of chemically cross-linked aromatic polyimide membrane a permeate gas composition containing a portion of at least one gas that permeated the membrane.    Y2a=compounds of formulae (IIIa), (IIIb) and (IIIc);    R1c=H or -COCH3.