The extension of the Dezert-Smarandache theory (DSmT) for the multi-class framework has a feasible computational complexity for various applications when the number of classes is limited or reduced typically two classes. In contrast, when the number of classes is large, the DSmT generates a high computational complexity. This paper proposes to investigate the effective use of the DSmT for multi-class classification in conjunction with the Support Vector Machines using the One-Against-All (OAA) implementation, which allows offering two advantages: firstly, it allows modeling the partial ignorance by including the complementary classes in the set of focal elements during the combination process and, secondly, it allows reducing drastically the number of focal elements using a supervised model by introducing exclusive constraints when classes are naturally and mutually exclusive. To illustrate the effective use of the DSmT for multi-class classification, two SVM-OAA implementations are combined according three steps: transformation of the SVM classifier outputs into posterior probabilities using a sigmoid technique of Platt, estimation of masses directly through the proposed model and combination of masses through the Proportional Conflict Redistribution (PCR6). To prove the effective use of the proposed framework, a case study is conducted on the handwritten digit recognition. Experimental results show that it is possible to reduce efficiently both the number of focal elements and the classification error rate.