I am having some trouble in understanding the meaning of the Exchange Theorem in page 45 of Cheney's "Introduction to Approximation Theory". The Haar condition for a subset $A$ of vectors in an $n$-dimensional (real) vector space is: every subset of $n$ elements of $A$ is linearly independent. It seems to me that if $B \subseteq A$ and $A$ satisfies the Haar condition, then $B$ also satisfies the Haar condition - because each subset of $n$ elements of $B$ is a subseteq of $n$ elements of $A$. The Exchange Theorem states that:
Let $\{A^0, \dots, A^{n+1}\}$ be a set of vectors in $n$ space satisfying the Haar condition. If $0$ lies in the convex hull of $\{A^0, \dots, A^n\}$, then there is $j \leq n$ such that this condition reamains true if $A^j$ is replaced by $A^{n+1}$.
I am having some confusion understanding this statement, possibly due to a trivial misunderstanding. First, we replace $A^j$ by $A^{n+1}$ in $\{A^0, \dots, A^{n+1}\}$ or in $\{A^0, \dots, A^n\}$? If so, as in each case we would end up with a subset of $\{A^0, \dots, A^{n+1}\}$, which should also satisfy the Haar condition. Could someone help me understand this result more clearly?