NOVELTY - An electrolytic membrane (1) comprises porous membrane substrate containing cross-linked polymer electrolyte having structural component containing repeating unit having aromatic hydrocarbon substituted by sulfonic acid; repeating unit having nitrogen-containing hetero ring compound residue, and its sulfate, hydrochloride, or organic sulfonate; and repeating unit having cross-linked group.    USE - For membrane electrode assembly used in fuel cell (claimed).    ADVANTAGE - The electrolytic membrane can be manufactured at low cost and causes less cross-over of methanol.    DETAILED DESCRIPTION - The electrolytic membrane comprises porous membrane substrate containing cross-linked polymer electrolyte having structural component of formula -(A)x-(B)y-(C)z- (1).    A=repeating unit having aromatic hydrocarbon substituted by sulfonic acid;    B=repeating unit having nitrogen-containing hetero ring compound residue, and its sulfate, hydrochloride, or organic sulfonate;    C=repeating unit having cross-linked group;and    X, Y, Z=mol fractions of respective repeating units.    It is given that 0.34less than or equal toXless than or equal to0.985, 0.005less than or equal toYless than or equal to0.49, 0.01less than or equal toZless than or equal to0.495, and Yless than or equal toX and Zless than or equal toX, provided that, in the repeating unit A, a ratio of the aromatic hydrocarbon substituted by sulfonic acid is 0.3-1, and the number of sulfonic acid in the aromatic hydrocarbon substituted by sulfonic acid is 1-3. INDEPENDENT CLAIMS are included for:    (1) a method for producing an electrolytic membrane, comprising impregnating a porous membrane substrate with a monomer mixture solution containing a monomer of Formula (2), a monomer of formula CH(R6)=CH2 (3), and a cross-linking agent of formula CH(R7-CH=CH2)=CH2 (4); polymerizing the monomer mixture by heating the porous membrane substrate impregnated with the monomer mixture; and sulfonating the polymerization product obtained in the polymerizing of the monomer mixture by using a sulfonating agent;    (2) a membrane electrode assembly comprising the electrolytic membrane having a fuel electrode joined onto its side and an oxidizer electrode joined onto its other side;    (3) a fuel cell comprising the membrane electrode assembly; and    (4) a method of operating the fuel cell comprising using the fuel cell and using a fuel containing methanol.    R1-R5=H, halo, optionally substituted, or hydrocarbon;    R6=optionally substituted nitrogen-containing five-membered group, or nitrogen-containing six-membered group;and    R7=optionally substituted organic group having valence of greater than or equal to2.    DESCRIPTION OF DRAWING(S) - The figure is a schematic diagram of the fuel cell.    Electrolytic membrane (1)    Fuel electrode (3)    Oxidizer electrode (4)