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Two constructive proofs on d-majorization and thermo-majorization areprovided. In the first part, we present a diagrammatic proof of the equivalencebetween d-majorization and the existence of a proper stochastic matrix. Weexplicitly construct the desired stochastic matrix by using a graphicalargument. In the second part, we present a constructive proof of theequivalence between the Gibbs-preserving map and thermal operation in classicalsystems. We construct the desired thermal operation and a heat bath whichemulates any Gibbs-preserving map with an arbitrary accuracy.