theorem Th70:
  for m being OperSymbol of C
  st the_result_sort_of m = a_Type & the_arity_of m = {}
  ex t st t = root-tree [m, the carrier of C] & t is pure
proof
  let m be OperSymbol of C such that
A1: the_result_sort_of m = a_Type and
A2: the_arity_of m = {};
  set X = MSVars C;
  root-tree [m, the carrier of C] in (the Sorts of Free(C, X)).a_Type by A1,A2,
MSAFREE3:5;
  then reconsider
  T = root-tree [m, the carrier of C] as expression of C, a_Type C
  by Th41;
  take T;
  thus T = root-tree [m, the carrier of C];
  given a,t such that
A3: T = (ast C)term(a,t);
  T = [ *, the carrier of C]-tree<*a,t*> by A3,Th46;
  then [ *, the carrier of C] = T.{} by TREES_4:def 4
    .= [m, the carrier of C] by TREES_4:3;
  then m = ast C by XTUPLE_0:1;
  hence contradiction by A2,Def9;
end;
