I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else.
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
Motivation is important too, but it’s not what I meant. A very simple example would be
Theorem: Every subspace Y of a second-countable topological space X is second-countable.
Reason: Intersecting each set in a basis for X with Y yields a basis for Y.
Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]
(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)
> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.