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 Th10:
  for o being OperSymbol of S st t.{} = [o,the carrier of S] ex a
  being ArgumentSeq of Sym(o,V) st t = [o,the carrier of S]-tree a
proof
  let o be OperSymbol of S such that
A1: t.{} = [o,the carrier of S];
  set X = V, G = DTConMSA X;
  reconsider t as Element of TS G;
  [o,the carrier of S] = Sym(o, X) by MSAFREE:def 9;
  then consider p being FinSequence of TS G such that
A2: t = Sym(o,X)-tree p and
A3: Sym(o,X) ==> roots p by A1,DTCONSTR:10;
  reconsider a = p as FinSequence of S-Terms V;
  a is SubtreeSeq of Sym(o,X) by A3,DTCONSTR:def 6;
  then reconsider a as ArgumentSeq of Sym(o,V) by Def2;
  take a;
  thus thesis by A2,MSAFREE:def 9;
end;
