reserve S for non void non empty ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S;
reserve A for MSAlgebra over S,
  t for Term of S,V;
reserve S for non void non empty ManySortedSign,
  A for non-empty MSAlgebra over S,
  V for Variables of A,
  t for c-Term of A,V,
  f for ManySortedFunction of V, the Sorts of A;

theorem Th35:
  for o being OperSymbol of S, p being ArgumentSeq of o,A,V for vt
being finite DecoratedTree st
   vt is_an_evaluation_of
    (Sym(o,(the Sorts of A) (\/) V)-tree p qua c-Term of A,V), f
  ex q being DTree-yielding FinSequence st len q
= len p & vt = (Den(o,A).roots q)-tree q & for i being Nat, t being c-Term of A
  ,V st i in dom p & t = p.i ex vt being finite DecoratedTree st vt = q.i & vt
  is_an_evaluation_of t,f
proof
  let o be OperSymbol of S, p be ArgumentSeq of o,A,V;
  let vt be finite DecoratedTree;
  assume
A1: vt is_an_evaluation_of (Sym(o,(the Sorts of A) (\/) V)-tree p qua
  c-Term of A,V), f;
  reconsider r = {} as empty Element of dom vt by TREES_1:22;
  consider x being set, q being DTree-yielding FinSequence such that
A2: vt = x-tree q by TREES_9:8;
A3: dom vt = tree doms q by A2,TREES_4:10;
  take q;
A4: len doms q = len q by TREES_3:38;
A5: Sym(o,(the Sorts of A) (\/) V) = [o,the carrier of S] by MSAFREE:def 9;
  then
A6: dom vt = dom ([o,the carrier of S]-tree p) by A1;
  then dom vt = tree doms p by TREES_4:10;
  then
A7: doms q = doms p by A3,TREES_3:50;
  hence len q = len p by A4,TREES_3:38;
  ([o,the carrier of S]-tree p).r = [o,the carrier of S] by TREES_4:def 4;
  then vt.r = Den(o, A).succ(vt,r) by A5,A1
    .= Den(o, A).roots q by A2,TREES_9:30;
  hence vt = (Den(o,A).roots q)-tree q by A2,TREES_4:def 4;
  let i be Nat, t be c-Term of A,V;
  assume that
A8: i in dom p and
A9: t = p.i;
  reconsider u = {} as Node of t by TREES_1:22;
  consider k being Element of NAT such that
A10: i = k+1 and
A11: k < len p by A8,Lm1;
  <*k*>^u = <*k*> by FINSEQ_1:34;
  then reconsider r = <*k*> as Node of vt by A6,A9,A10,A11,TREES_4:11;
  take e = vt|r;
  len doms p = len p by TREES_3:38;
  hence e = q.i by A2,A7,A4,A10,A11,TREES_4:def 4;
  reconsider r1 = r as Node of [o,the carrier of S]-tree p by A5,A1;
  t = ([o,the carrier of S]-tree p)|r1 by A9,A10,A11,TREES_4:def 4;
  hence thesis by A5,A1,Th34;
end;
