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;

theorem Th3:
  len v1 = k+1 implies ex v2,x st v1 = v2^<*x*> & len v2 = k
proof
  assume
A1: len v1 = k+1;
  reconsider v2 = v1|Seg k as FinSequence by FINSEQ_1:15;
 v2 is_a_prefix_of v1;
  then consider v such that
A2: v1 = v2^v by TREES_1:1;
  take v2, v.1;
A3: k <= k+1 by NAT_1:11;
then  len v2 = k by A1,FINSEQ_1:17;
then  k + len v = k+1 by A1,A2,FINSEQ_1:22;
  hence thesis by A1,A2,A3,FINSEQ_1:17,40;
end;
