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;
reserve s,t for FinSequence;

theorem
  for T being Tree, t being Element of T holds
  t in Leaves T iff not t^<*0*> in T
proof
  let T be Tree, t be Element of T;
  hereby
    assume
A1: t in Leaves T;
 t is_a_proper_prefix_of t^<*0*> by Th6;
    hence not t^<*0*> in T by A1,Def5;
  end;
  assume that
A2: not t^<*0*> in T and
A3: not t in Leaves T;
  consider q being FinSequence of NAT such that
A4: q in T and
A5: t is_a_proper_prefix_of q by A3,Def5;
 t is_a_prefix_of q by A5;
  then consider r being FinSequence such that
A6: q = t^r by Th1;
  reconsider r as FinSequence of NAT by A6,FINSEQ_1:36;
 len q = len t+len r by A6,FINSEQ_1:22;
 then len r <> 0 by A5,Th5;
 then r <> {};
 then consider p being FinSequence of NAT,
   x being Element of NAT such that
A7: r = <*x*>^p by FINSEQ_2:130;
 q = t^<*x*>^p by A6,A7,FINSEQ_1:32;
 then  t^<*x*> in T by A4,Th20;
  hence contradiction by A2,Def3;
end;
