reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;

theorem Th22:
  v1 in C & v2 in C & v is_a_prefix_of v2 implies
  v1 in ProperPrefixes v or v is_a_prefix_of v1
proof
  assume that
A1: v1 in C & v2 in C and
A2: v is_a_prefix_of v2;
 v1 in ProperPrefixes v2 or v2 is_a_prefix_of v1 by A1,Th21;
then  v1 is_a_proper_prefix_of v2 or v is_a_prefix_of v1
  by A2,TREES_1:12;
then  v,v1 are_c=-comparable by A2,Th2;
then  v is_a_proper_prefix_of v1 or v = v1 or v1 is_a_proper_prefix_of v;
  hence thesis by TREES_1:12;
end;
