TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Process: The polymeric compound is formed by grafting a monomer (M) which is a compound having cationic group containing a nitrogen atom in the molecule, with respect to a polymeric base material, and reacting sultone with respect to the nitrogen atom. Alternatively, the monomer (M) is a compound having an aromatic ring and a cationic group in the molecule. A sulfo group is introduced into the aromatic ring after grafting by sulfonation. The molecules of polymeric base material are cross-linked by a cross-linking agent.Preferred Compounds: The anionic and the cationic groups exist preferably in the same polymeric molecule. The anionic compound is chosen from sulfuric acid, phosphoric acid, hydrochloric acid or trifluoromethane sulfonic acid. The polymeric material contains a fluorine atom. The cationic group contains nitrogen, and is preferably a pyridine derivative. The polymer compound (having anionic and cationic groups on the same molecule) having a substituent of formula (1) in the side chain, is preferably a compound having formula (2) or (3) as main structure. EXAMPLE - A resin film containing ethylene tetrafluoroethylene copolymer and having thickness of 25 microns, was irradiated with electron beam of 400 kGy. The film was immersed in a solution containing 4-vinylpyridine:ethylene glycol divinylether:toluene in the volume ratio of 1:0.18:1. Polymerization was carried out 70degreesC for 3 hours with nitrogen bubbling. The polymer which was not grafted was removed. The solution containing propane sultone dissolved in N-methyl pyrrolidone, was reacted with the formed grafted polymer at 70degreesC for 12 hours, and a polymer thin film was produced. The polymer compound in the formed polymeric film was represented as formula (2), where n =3, p = 4500, q = 4500 and r = 20. The polymeric thin film was immersed in 32 mass% sulfuric acid aqueous solution at 25degreesC for 24 hours so that the cationic group in the film was neutralized. The film was then washed, and a polymeric electrolyte film having moisture content of 24% and excellent tensile strength was obtained.