TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Agent: The cross-linking agent is di N-methylol compound, dicarboxylic acid compound, dicarboxylic acid chloride compound, bis halogen compound, bis epoxide compound, bis aziridine compound, diisocyanate compound, or 1 type or 2 or more types chosen from boric acid group compound.Preferred Method: An acid is further added in process (3) after process (2).    TECHNOLOGY FOCUS - POLYMERS - Preferred Property: The polymeric electrolyte film has ion exchange capacity of 0.1-5 meq/g. Preferred Definitions:    m,n = 1;    R1,R2 = methyl; and    R3 = ethylene, propylene or butylene.    EXAMPLE - Polyarylene ether sulfone (in g) containing hydroxyl group (6.6 mmol) was dissolved in tetrahydrofuran (360 ml). Subsequently, methanol solution (144 ml) containing sodium methoxide (0.03 mmol/l) was added at room temperature and stirred for 1 hour. Tetrahydrofuran solution (16.4 ml) containing 1,3-propane sulfone (1.2 mol/l) was added and stirred at room temperature for 20 hours. A solid was obtained by distilling the solvent. The solid was vacuum dried at 110 degrees C for 4 hours, and was washed with methylene chloride, hydrochloric acid (0.1 mol/l) and water for 24 hours to obtain polyarylene ether sulfone having 1.4 g of sulfopropoxy group and hydroxyl group. The polyarylene ether sulfone (1) containing sulfo propoxy group (1.8 mmol) and hydroxyl group (0.99 mmol) was mixed with dimethyl sulfoxide (15 ml) and stirred to form a homogenous solution. Methanol solution (3) containing sodium methoxide (1 mol/l) was added and stirred at room temperature for 1 hour. Terephthal acid chloride (0.57) was added and stirred for 5 minutes. The pressure in the system was reduced to 1 x 10 to the power of -4 MPa at 80 degrees C for 16 hours, and solvent was distilled to obtain a cross-linked polyarylene ether sulfone. The sulfone was washed with water, and dipped in boiling water for 30 minutes to obtain cross-linked polyarylene ether sulfone having excellent water resistance. The cross-linked polyarylene ether sulfone was used for forming polymeric electrolyte film.