reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;

theorem
  for t being finite DecoratedTree, x being set, p being DTree-yielding
FinSequence st t = x-tree p for n being empty Element of dom t holds succ(t,n)
  = roots p
proof
  let t be finite DecoratedTree, x be set;
  let p be DTree-yielding FinSequence such that
A1: t = x-tree p;
  let n be empty Element of dom t;
A2: len doms p = len p by TREES_3:38;
  now
    let x be object;
    assume x in dom doms p;
    then consider i being Nat such that
A3: x = i+1 & i < len p by A2,Lm1;
    reconsider i as Element of NAT by ORDINAL1:def 12;
A4: p.x = t|<*i*> by A1,A3,TREES_4:def 4;
    n in dom (t|<*i*>) & <*i*>^n = <*i*> by FINSEQ_1:34,TREES_1:22;
    then reconsider ii = <*i*> as Node of t by A1,A3,A4,TREES_4:11;
    x in dom p by A3,Lm3;
    then (doms p).x = dom (t|ii) by A4,FUNCT_6:22;
    hence (doms p).x is finite Tree;
  end;
  then reconsider dp = doms p as FinTree-yielding FinSequence by TREES_3:23;
A5: dom t = tree dp by A1,TREES_4:10;
A6: ex q being Element of dom t st q = n & succ(t,n) = t*(q succ) by Def6;
  rng (n succ) c= dom t;
  then dom succ(t,n) = dom (n succ) by A6,RELAT_1:27;
  then
A7: len succ(t,n) = len (n succ) by FINSEQ_3:29;
  then
A8: len succ(t,n) = card succ n by Def5
    .= len p by A2,A5,Th29;
A9: now
    let i be Nat;
    assume
A10: i < len p;
    reconsider ii=i as Element of NAT by ORDINAL1:def 12;
    i+1 in dom p by Lm3,A10;
    then
A11: {} in (dom t)|<*ii*> & ex T being DecoratedTree st T = p.(i+1) & (
    roots p).(i +1) = T.{} by TREES_1:22,TREES_3:def 18;
    p.(i+1) = t|<*ii*> by A1,A10,TREES_4:def 4;
    then
A12: (roots p).(i+1) = t.(<*i*>^{}) by A11,TREES_1:22,TREES_2:def 10;
    i+1 in dom succ(t,n) by A8,A10,Lm3;
    then succ(t,n).(i+1) = t.((n succ).(i+1)) by A6,FUNCT_1:12
      .= t.(n^<*i*>) by A7,A8,A10,Def5
      .= t.<*i*> by FINSEQ_1:34;
    hence succ(t,n).(i+1) = (roots p).(i+1) by A12,FINSEQ_1:34;
  end;
  dom roots p = dom p by TREES_3:def 18;
  then len roots p = len p by FINSEQ_3:29;
  hence thesis by A8,A9,Lm2;
end;
