All Horses have the same colour

We will apply Mathematical Induction to the following statement

All horses have the same color pattern

1. Induction Hypothesis: Any set of $n$ horses $\{h_1,h_2,...h_n\}$ have the same color pattern.
2. Base case: For $n=1$ we have a single horse, so there can be only one color pattern.
3. Inductive Step: Assume that the statement is true for $n$. Take the set $\{h_1,h_2,...,h_n,h_{n+1}\}$ with $n+1$ horses. By our hypothesis the set of horses $\{h_1,...,h_n\}$ has a single color pattern, and the set $\{h_2,...,h_n,h_{n+1}\}$ also has a single color pattern. Since this must be the color pattern of their intersection set $\{h_2,...h_n\}$, then the entire set of $n+1$ horses has the same color.

Conclusion

All three elements of mathematical induction are fulfilled. Can you spot the logical error?