Use of a polyacrylic acid cross-linked by a cross-linking agent selected from the compounds described in the following general formulae, and to and the polymer described in the following general formula, provided that the one which includes a functional group-containing vinylidene fluoride-based polymer is excluded, in a binder for a lithium cell; wherein k represents an integer of 1 to 6, wherein R 25 and R 26 each independently represent a hydrogen atom or a methyl group, R 20 represents an alkylene group having 1 to 20 carbon atoms, a group represented by the following general formula, wherein R 17 represents an alkylene group having 1 to 6 carbon atoms, r represents an integer of 1 to 6, or the group represented by the following general formula wherein R 18 and R 19 each independently represent an alkylene group having 1 to 6 carbon atoms, t represents an integer of 1 to 12, wherein R 7 to R 10 each independently represent an alkylene group having 1 to 6 carbon atoms, 1 represents an integer of 1 to 6, m represents an integer of 0 to 6, v1 and v2 each independently represent an integer of 0 or 1, wherein R 31 to R 38 each independently represent a hydrogen atom, a vinyl group or a vinylketone group, and at least two or more groups thereof are a vinyl group or a vinylketone group, wherein R 12 to R 14 each independently represent an alkylene group having carbon 1 to 6 atoms, wherein Ar 1 represents an aryl group having 6 to 10 carbon atoms, R 15 represents an alkylene group having 1 to 6 carbon atoms, q represents an integer of 2 to 4, wherein Ar 2 and Ar 3 each independently represent an arylene group having 6 to 10 carbon atoms, R 16 represents an alkylene group having 1 to 6 carbon atoms, wherein Ar 4 represents an arylene group having 6 to 10 carbon atoms, wherein, R 11 represents an alkylene group having 1 to 6 carbon atoms, wherein R 21 represents an alkylene group having 1 to 6 carbon atoms which has or does not have a substituent, an arylene group having 6 to 10 carbon atoms which has or does not have a substituent, or a group represented by the following general formula [12-A] wherein R 35 represents an alkyl group having 1 to 6 carbon atoms, R 36 represents an alkylene group having 1 to 6 carbon atoms, Ar 5 represents an arylene group having 6 to 10 carbon atoms which may have b pieces of R 35 as substituents, b represents an integer of 0 to 4, or the group represented by the following general formula [12-B] wherein R 37 represents an alkylene group having 1 to 6 carbon atoms, R 35, R 36, Ar 5 and b are the same as described above, wherein R 22 each independently represents an alkyl group having 1 to 6 carbon atoms which has or does not have a substituent, an aryl group having 6 to 10 carbon atoms which has or does not have a substituent, or the group represented by the following general formula [13-A] wherein R 40 represents an alkyl group having 1 to 6 carbon atoms, R 41 represents an alkylene group having 1 to 6 carbon atoms, Ar 6 represents an arylene group having 6 to 10 carbon atoms which have c pieces of R 40 as substituents, c represents an integer of 0 to 5, wherein R 23 represents an alkylene group having 1 to 6 carbon atoms which has or does not have a substituent, an arylene group having 6 to 10 carbon atoms which has or does not have a substituent, or a group represented by the following general formula [14-A] or [14-B] wherein R 30 represents an alkyl group having 1 to 6 carbon atoms, R 31 represents an alkylene group having 1 to 6 carbon atoms, Ar 7 represents an arylene group having 6 to 10 carbon atoms which may have a pieces of R 30 as substituents, a represents an integer of 0 to 4, or the group following general formula [14-C] wherein R 32 represents an alkylene group having 1 to 6 carbon atoms, R 30, R 31, Ar 7, and a are the same as described above, and s represents an integer of 10 to 10000.