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The uniform parabolicity condition 0 < k(1) <= k(alpha)(u) <= k(2), alpha = 1, 2 is assumed to be fulfilled for the sign alternating solution u(x, t) is an element of D (u) only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle not only two-sided estimates for the approximate solution y but its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives partial differential u partial differential xi have discontinuities of the first kind in the neighbourhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. 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