In the proof of uniqueness of a polygon with given side lengths for euclidean geometry, we showed that, if A(0),...,A(n) lay in order on a euclidean circle K and the A(i)A(i+1) were fixed, then the length A(0)A(n) increased with the radius of K. Although the same strategy works in hyperbolic geometry, the proofs are harder.
Earlier, we looked at an ilines L intersecting the disk D in K. Then
We also used arguments which involved varying a euclidean circle In our proof, we will need some results on saccheri quadrilaterals.

a monotonicity theorem Suppose that (1) K is the intersection of an iline with the disk, (2) l(1),..,l(m) are fixed positive numbers, and (3) A(0),..A(m) lie in order on the minor (or finite) arc A(0)A(m) of K such that l(i) = d(A(i),A(i+1)), i = 1,..,m1. Then, as the size of K increases, L = d(A(0),A(m)) increases continuously, and tends to l(1)+..+l(m).
We are now ready to prove the existence of convex hyperbolic polygons under the 