reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;
reserve C for initialized ConstructorSignature,
  s for SortSymbol of C,
  o for OperSymbol of C,
  c for constructor OperSymbol of C;
reserve a,b for expression of C, an_Adj C;
reserve t, t1,t2 for expression of C, a_Type C;
reserve p for FinSequence of QuasiTerms C;
reserve e for expression of C;
reserve a,a9 for expression of C, an_Adj C;

theorem Th71:
  for v being OperSymbol of C
  st the_result_sort_of v = an_Adj & the_arity_of v = {}
  ex a st a = root-tree [v, the carrier of C] & a is positive
proof
  let m be OperSymbol of C such that
A1: the_result_sort_of m = an_Adj and
A2: the_arity_of m = {};
  set X = MSVars C;
  root-tree [m, the carrier of C] in (the Sorts of Free(C, X)).an_Adj
  by A1,A2,MSAFREE3:5;
  then reconsider
  T = root-tree [m, the carrier of C] as expression of C, an_Adj C
  by Th41;
  take T;
  thus T = root-tree [m, the carrier of C];
  given a being expression of C, an_Adj C such that
A3: T = (non_op C)term a;
  T = [non_op, the carrier of C]-tree<*a*> by A3,Th43;
  then [non_op, the carrier of C] = T.{} by TREES_4:def 4
    .= [m, the carrier of C] by TREES_4:3;
  then m = non_op by XTUPLE_0:1;
  hence contradiction by A2,Def9;
end;
