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;

theorem Th1:
  for o being OperSymbol of S, a being FinSequence holds [o,the
carrier of S]-tree a in S-Terms V & a is DTree-yielding iff a is ArgumentSeq of
  Sym(o,V)
proof
  let o be OperSymbol of S, a be FinSequence;
A1: [o,the carrier of S] = Sym(o, V) by MSAFREE:def 9;
A2: S-Terms V = TS DTConMSA V;
  hereby
    assume [o,the carrier of S]-tree a in S-Terms V;
    then reconsider
    t = [o,the carrier of S]-tree a as Element of TS DTConMSA V;
    assume
A3: a is DTree-yielding;
    then t.{} = [o,the carrier of S] by TREES_4:def 4;
    then consider p being FinSequence of TS DTConMSA V such that
A4: t = Sym(o, V)-tree p and
A5: Sym(o, V) ==> roots p by A1,DTCONSTR:10;
    a = p by A3,A4,TREES_4:15;
    then a is SubtreeSeq of Sym(o, V) by A5,DTCONSTR:def 6;
    hence a is ArgumentSeq of Sym(o,V) by A2,Def2;
  end;
  assume a is ArgumentSeq of Sym(o,V);
  then reconsider a as ArgumentSeq of Sym(o,V);
  reconsider p = a as FinSequence of TS DTConMSA V by Def1;
  p is SubtreeSeq of Sym(o, V) by Def2;
  then Sym(o, V) ==> roots p by DTCONSTR:def 6;
  hence thesis by A1,DTCONSTR:def 1;
end;
