In The Alchemist, are the king, the crystal merchant, and the alchemist the same person or spirit?

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It is important to realise that these three characters are very distinct and different in the roles that they perform in the novel. The two who are most similar are the King, Melchizedek, and the Alchemist, yet even these two are clearly distinct. Melchizedek, although he appears to Santiago in flesh and blood, is some kind of spiritual being whose role is to encourage all humans to continue pursuing their Personal Legend, even during times of doubt. It is he who tells Santiago about Personal Legends and gives him the resources he needs to be able to follow his own Personal Legend. The Alchemist is likewise another important person who helps Santiago, as he teaches Santiago so much about what he needs to do in order to achieve his own Personal Legend, and particularly about the importance of overcoming his fear. It is the Crystal Merchant, however, who is the most obviously different out of all three of these individuals, for it is he who represents the failure or fear to pursue one's Personal Legend. Note what he says to Santiago:

I don't want anything else in life. But you are forcing me to look at wealth and at horizons that I have never known. Now that I have seen them, and now that I see how immense my possibilities are, I'm going to feel worse than I did before you arrived. Because I know the things I should be able to accomplish, and I don't want to do so.

The Crystal Merchant deliberately rejects his own Personal Legend to go on a pilgrimage to Mecca, because he fears that he will have nothing to live for afterwards. Even though Santiago teaches him about the importance of taking risks, it is obvious that the Crystal Merchant is used to represent a salutory lesson of the frustrations and guilt that humans experience if they do not pursue their own Personal Legend. Thus each of these three characters are distinct and separate, and are used in very different ways to reinforce the central themes of the novel.