In "Republic" lines 439a to 441d, on what grounds does Plato argue that the spirited part is distinct from the other two parts of the soul?
In Resp. 439-441, Socrates argues for the essentially tripartite nature of the soul inductively, looking at examples of people behaving in ways that cannot be accounted for solely by either intellect or bodily desires. Thumos cannot be part of the appetitive part of the soul because it acts against our desires as when we feel compelled to look at something we find repugnant, but neither can it be part of the rational soul as it causes us to act impulsively in ways that are not bodily desires but will go against the mandates of reason. Like righteous indignation in Aristotle, if properly trained, it can be a positive moral force.
One thing you need to note as you read the Republic is that Plato sets forth at the beginning of the dialogue that it is difficult to understand the soul, and thus to study the soul, he will examine the city as metaphor for the soul because it is a form of the soul writ large. Thus consider the class of helpers in the city as akin to thumos.