reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;
reserve T for DecoratedTree,
  p for FinSequence of NAT;
reserve T for finite-branching DecoratedTree,
  t for Element of dom T,
  x for FinSequence,
  n, m for Nat;
reserve x, x9 for Element of dom T,
  y9 for set;

theorem
  y9 in rng succ(T,x) implies ex x9 st y9 = T.x9 & x9 in succ x
proof
  consider k be Nat such that
A1: dom succ(T,x) = Seg k by FINSEQ_1:def 2;
  assume y9 in rng succ(T,x);
  then consider n9 being object such that
A2: n9 in dom succ(T,x) and
A3: y9 = (succ(T,x)).n9 by FUNCT_1:def 3;
  Seg k = { m where m is Nat : 1 <= m & m <= k } by FINSEQ_1:def 1;
  then consider m9 being Nat such that
A4: n9 = m9 and
A5: 1 <= m9 and
  m9 <= k by A2,A1;
  m9 <> 0 by A5;
  then consider n being Nat such that
A6: n+1 = m9 by NAT_1:6;
  reconsider n as Nat;
  n+1 in dom (x succ) by A2,A4,A6,Th38;
  then x^<*n*> in dom T by Th39;
  then consider x9 such that
A7: x9 = x^<*n*>;
A8: x9 in succ x by A7;
  y9 = T.x9 by A3,A4,A6,A7,Th40;
  hence thesis by A8;
end;
