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 Th21:
  v1 in C & v2 in C implies v1 in ProperPrefixes v2 or v2 is_a_prefix_of v1
proof
  assume
A1: v1 in C & v2 in C;
  then reconsider p = v1, q = v2 as Element of W;
  assume
A2: not v1 in ProperPrefixes v2;
A3: p,q are_c=-comparable by A1,Def3;
A4: not v1 is_a_proper_prefix_of v2 by A2,TREES_1:12;
 v1 is_a_prefix_of v2 or v2 is_a_prefix_of v1 by A3;
  hence thesis by A4;
end;
