> We will show, by induction, that [for all sets] of n horses, every horse in [each] set has the same color.
> Now [assume that] for all sets of n horses, every horse in [each] set has the same color.
QED.
This is just tautology. No further analysis needed, really.
(If your proof includes your hypothesis as an assumption, then it must be a proof by contradiction.)
EDIT: Before you refute, read again carefully. The assumption _is_ the hypothesis. It is not existentially quantified or set in the base case. It is the entire, universally quantified hypothesis.
> Now [assume that] for all sets of n horses, every horse in [each] set has the same color.
QED.
This is just tautology. No further analysis needed, really.
(If your proof includes your hypothesis as an assumption, then it must be a proof by contradiction.)
EDIT: Before you refute, read again carefully. The assumption _is_ the hypothesis. It is not existentially quantified or set in the base case. It is the entire, universally quantified hypothesis.