TECHNOLOGY FOCUS - POLYMERS - Preferred Polymer: The polymer has a molecular weight of 500-500,000 and is an oxygen-containing polymer having at least one of a carbonyl group and an ether group, preferably of formulae -(R1-COO-R2)k- (I), -(R3-COO-R4-OCO-R5)m- (II), -(R6-COO-R7-CO-R8)n- (III) or -(R9-COO-R10-O-R11)g- (IV);R1-11 = hydrocarbon groups.Preferred Battery: The negative electrode comprises a carbonaceous material. The positive electrode comprises a lithium-transition metal double oxide. The non-aqueous solvent of the non-aqueous electrolytic solution comprises a carbonate-based solvent. The electrolytic salt of the non aqueous electrolytic solution is a lithium salt.Preferred Method: A charging voltage during charging and discharging under overcharge conditions is set at 4.2 V or more. EXAMPLE - 90 parts weight (pts.wt.) of non-graphitising carbon material and 10 pts.wt. of polyvinylidene fluoride (PVDF) as a binder were mixed and dispersed in N-methyl-2-pyrrolidone as a solvent to prepare a negative electrode mixture slurry. A strip copper foil having a thickness of 10 mum was prepared as a negative charge collector. The negative electrode mixture slurry was uniformly applied to both surfaces of the negative charge collector and dried, and then, compression molding was performed at constant pressure to fabricate a strip negative electrode. 91 pts.wt. of lithium cobaltate, 6 pts.wt. of carbon powder and 3 pts.wt. of polyvinylidene fluoride as a binder were mixed and dispersed in N-methyl-2-pyrrolidone as a solvent to prepare a positive electrode mixture slurry. A strip aluminum foil having a thickness of 20 mum was prepared as a positive charge collector. The positive electrode mixture slurry was uniformly applied to both surfaces of the positive charge collector and dried, and then, compression molding was performed to fabricate a strip positive electrode. The strip negative electrode and the strip positive electrode were laminated with separators composed of a microporous polypropylene film, in the order of the negative electrode, the separator, the positive electrode, and the separator, and were wound up many times to form a wound electrode member. This was placed in an iron battery can plated with nickel. The top and bottom of the coiled-type electrode were provided with an insulator, an aluminum positive electrode lead drawn from the positive charge collector was welded to a sheet for breaking a current, and a nickel negative electrode drawn from the negative charge collector was welded to the battery can. Into a mixed solvent of equivolume propylene carbonate and dimethyl carbonate, 3 wt.% of polyester having formula -(CH2-COO-CH2)100- and 1 mol/l of LiPF6 were dissolved to prepare a non-aqueous electrolytic solution. The nonaqueous electrolytic solution was poured into the battery can. A battery cap was fixed on the sheet for breaking a current by caulking joints with an insulated sealing gasket applied with asphalt being interposed, and outrightness within the battery was maintained, and thus a cylindrical non aqueous electrolytic secondary battery was fabricated.