Let $a,b$ be affine combinations of points from a set $S$. Then is the affine combination of $a,b$ also an affine combination of points from $S$?

Question

Let A be an affine space, $a,b$ affine combinations of points from a finite subset $S$ of A. Then is the affine combination of $a,b$ also an affine combination of points from $S$?

I found it hard to show because it involves two affine combinations and things get quite complicatd.

Answer 1

If $a=c_i s^i$ and $b=d_i s^i$ (both sums) then $\lambda a +\mu b=(\lambda c_i+\mu d_i)s^i$.