reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));

theorem
  for a being set, w being DTree-yielding FinSequence holds
  dom (a-tree w) = {{}} \/ union {<*i*>^^dom (w.(i+1)): i < len w}
  proof let a be set;
    let w be DTree-yielding FinSequence;
    set A = {<*i*>^^dom (w.(i+1)): i < len w};
    thus dom (a-tree w) c= {{}} \/ union A
    proof
      let b; reconsider x = b as set by TARSKI:1;
      assume b in dom (a-tree w);
      then per cases by TREES_4:11;
      suppose x = {};
        then x in {{}} by TARSKI:def 1;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose ex i st ex T being DecoratedTree, q being Node of T st
        i < len w & T = w.(i+1) & x = <*i*>^q;
        then consider i being Nat, T being DecoratedTree, q being Node of T
        such that
A1:     i < len w & T = w.(i+1) & x = <*i*>^q;
        x in <*i*>^^dom T & <*i*>^^dom T in A by A1;
        then x in union A by TARSKI:def 4;
        hence thesis by XBOOLE_0:def 3;
      end;
    end;
    let b; reconsider x = b as set by TARSKI:1;
    assume b in {{}} \/ union A;
    then per cases by XBOOLE_0:def 3;
    suppose x in {{}};
      then x = {};
      hence thesis by TREES_4:11;
    end;
    suppose x in union A;
      then consider I such that
A2:   x in I & I in A by TARSKI:def 4;
      consider i such that
A3:   I = <*i*>^^dom (w.(i+1)) & i < len w by A2;
      consider q being Element of dom (w.(i+1)) such that
A4:   x = <*i*>^q & q in dom (w.(i+1)) by A2,A3;
      1 <= i+1 <= len w by A3,NAT_1:11,13;
      then i+1 in dom w by FINSEQ_3:25;
      then w.(i+1) in rng w by FUNCT_1:def 3;
      then reconsider T = w.(i+1) as DecoratedTree;
      q in dom T;
      hence thesis by A3,A4,TREES_4:11;
    end;
  end;
