reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;
reserve T,T1 for Tree;
reserve fT,fT1 for finite Tree;
reserve t for Element of T;
reserve w for FinSequence;
reserve t1,t2 for Element of T;

theorem
  for t being Element of fT holds height(fT|t) <= height fT
proof
  let t be Element of fT;
  consider p such that
A1: p in fT|t and
A2: len p = height(fT|t) by Def12;
 t^p in fT by A1,Def6;
then A3: len(t^p) <= height fT by Def12;
   len(t^p) = len t + len p & len p <= len p + len t by FINSEQ_1:22,NAT_1:11;
  hence thesis by A2,A3,XXREAL_0:2;
end;
