reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th51:
  for p1,p2 being Tree-yielding FinSequence, T being Tree holds
  p in T iff <*len p1*>^p in tree(p1^<*T*>^p2)
proof
  let p1,p2 be Tree-yielding FinSequence, T be Tree;
A1: len (p1^<*T*>^p2) = len (p1^<*T*>) + len p2 by FINSEQ_1:22;
A2: len (p1^<*T*>) = len p1 + len <*T*> by FINSEQ_1:22;
A3: len <*T*> = 1 by FINSEQ_1:40;
A4: len p1+1+len p2 = (len p1+len p2)+1;
  len p1 <= len p1+len p2 by NAT_1:11;
  then
A5: len p1 < len (p1^<*T*>^p2) by A1,A2,A3,A4,NAT_1:13;
  len p1+1 >= 1 by NAT_1:11;
  then len p1+1 in dom (p1^<*T*>) by A2,A3,FINSEQ_3:25;
  then
A6: (p1^<*T*>^p2).(len p1+1) = (p1^<*T*>).(len p1+1) by FINSEQ_1:def 7
    .= T by FINSEQ_1:42;
  hence p in T implies <*len p1*>^p in tree(p1^<*T*>^p2) by A5,Def15;
  assume <*len p1*>^p in tree(p1^<*T*>^p2);
  then consider n,q such that
  n < len (p1^<*T*>^p2) and
A7: q in (p1^<*T*>^p2).(n+1) and
A8: <*len p1*>^p = <*n*>^q by Def15;
A9: (<*len p1*>^p).1 = len p1 by FINSEQ_1:41;
  (<*n*>^q).1 = n by FINSEQ_1:41;
  hence thesis by A6,A7,A8,A9,FINSEQ_1:33;
end;
