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With the purpose of interpreting modern satellite observations of successively increasing the brightness of individual coronal loops in solar flares, we solved the problem of the stability of small longitudinal perturbations of a homogeneous reconnecting current layer in the MHD approximation. The suppression of the plasma thermal conductivity by the magnetic field inside the current layer provides an instability. The instability increases in a radiative cooling time scale of the plasma in the linear phase. A periodic structure of hot and cold fibers arranged transversely to the direction of the electric current are formed as a result of the instability. The proposed mechanism of the thermal instability can be useful for an explanation of the consistent increase in the brightness of individual coronal loops in solar flares.