The present invention relates to novel optionally substituted 8-cyano-1-cyclo-propyl-7-(2,8-diazabicyclo[4.3.0]nonan-8-yl)-6-fluoro-1,4-dihydro-4-oxo-3-quinolinecarboxylic acids and their derivatives, of the general formula (I) in which R1 represents hydrogen, C1-C4-alkyl which is optionally substituted by hydroxyl, methoxy, amino, methylamino or dimethylamino, or (5-methyl-2-oxo-1,3-dioxol-4-yl)methyl, R2 represents hydrogen, benzyl, C1-C3-alkyl, (5-methyl-2-oxo-1,3-dioxol-4-yl)methyl, radicals having the structures —CH═CH—COOR3, —CH2CH2COOR3, —CH2CH2CN, —CH2CH2COCH3 or —CH2COCH3, in which R3 represents methyl or ethyl, or a radical of the general structure R4—(NH—CHR5—CO)n—, in which R4 represents hydrogen, C1-C3-alkyl or the radical —COO-tert-butyl, R5 represents hydrogen, C1-C4-alkyl, hydroxyalkyl, aminoalkyl, thioalkyl, carboxyalkyl or benzyl and n is 1 or 2, and Y is oxygen or sulfur, the process for their preparation and their use in antibacterial compositions.