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 Th25:
  p is_a_prefix_of q & q in B implies p in B
proof
  assume p is_a_prefix_of q;
then  p is_a_proper_prefix_of q or p = q;
then A1: p in ProperPrefixes q or p = q by TREES_1:def 2;
  assume
A2: q in B;
then  ProperPrefixes q c= B by Def7;
  hence thesis by A1,A2;
end;
