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;
reserve t for c-Term of A,V;

theorem
  for o being OperSymbol of S, p being ArgumentSeq of o,A,V for q being
FinSequence st len q = len p & for i being Nat st i in dom p for t being c-Term
  of A,V st t = p.i holds q.i = t@f
 holds (Sym(o,(the Sorts of A) (\/) V)-tree p
  qua c-Term of A,V)@f = Den(o,A).q
proof
  let o be OperSymbol of S, p be ArgumentSeq of o,A,V;
  let q be FinSequence;
  assume that
A1: len q = len p and
A2: for i being Nat st i in dom p for t being c-Term of A,V st t = p.i
  holds q.i = t@f;
  consider vt being finite DecoratedTree such that
A3: vt is_an_evaluation_of Sym(o,(the Sorts of A) (\/) V)-tree p, f by Th36;
  consider r being DTree-yielding FinSequence such that
A4: len r = len p and
A5: vt = (Den(o,A).roots r)-tree r and
A6: 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 = r.i & vt is_an_evaluation_of t,f by A3,Th35;
  now
    thus
A7: dom p = dom p & dom q = dom p & dom r = dom p by A1,A4,FINSEQ_3:29;
    let i be Element of NAT;
    assume
A8: i in dom r;
    then reconsider t = p.i as c-Term of A,V by A7,Th22;
    consider vt being finite DecoratedTree such that
A9: vt = r.i and
A10: vt is_an_evaluation_of t,f by A6,A7,A8;
    reconsider T = vt as DecoratedTree;
    take T;
    thus T = r.i by A9;
    thus q.i = t@f by A2,A7,A8
      .= T.{} by A10,Th39;
  end;
  then q = roots r by TREES_3:def 18;
  hence
  (Sym(o,(the Sorts of A) (\/) V)-tree p qua c-Term of A,V)@f = (((Den(o,A)
  ).q)-tree r).{} by A3,A5,Th39
    .= Den(o,A).q by TREES_4:def 4;
end;
