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 Th47:
  for t being Element of fT st t <> {} holds height(fT|t) < height fT
proof
  let t be Element of fT;
  assume t <> {};
then A1: 0+1 <= len t by NAT_1:13;
  consider p such that
A2: p in fT|t and
A3: len p = height(fT|t) by Def12;
 t^p in fT by A2,Def6;
then A4: len(t^p) <= height fT by Def12;
   len(t^p) = len t + len p & len p + 1 <= len t + len p by A1,FINSEQ_1:22
,XREAL_1:7;
then  height(fT|t)+1 <= height fT by A3,A4,XXREAL_0:2;
  hence thesis by NAT_1:13;
end;
