We proceed by induction. Let S={n in N | in any set of n horses, all members are the same color}. Since in any set of 1 horse, all members are the same color, we know that 1 is in S. For the induction step, let k be an element of N and assume that k is an element of S. We must show that k+1 is an element of S, ie in any set of k+1 horses, all members are the same color. Let A be a set of k+1 horses, call them H1,...,H(k+1). We will show that they all have the same color. Since {H2,...,H(k+1)} is a set of k horses, and k is in S, we know that H2,...,H(k+1) all have the same color. Since {H1,...,Hk} is a set of k horses, and k is in S, we also know that H1,...,Hk all have the same color. Thus H1 and H2 are the same color, and hence H1,...,H(k+1) are the same color. Therefore, k+1 is an element of S. Therefore S=N, ie all horses are the same color.
Match the statement to the proof that either proves or disproves it.