reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));

theorem Th11:
  T deg<= 0 = the set of all x-term
  proof
    thus T deg<= 0 c= the set of all x-term
    proof
      let a; assume a in T deg<= 0;
      then consider r such that
A1:   a = r & deg r <= 0;
A2:   deg @r = deg r = 0 by A1;
      reconsider t = r as Element of Free(S,X) by MSAFREE4:39;
      (ex s,x st t = x-term) or ex o,p st t = o-term p by Th16;
      hence thesis by A1,A2,Th22;
    end;
    let a; assume a in the set of all x-term;
    then consider s,x such that
A3: a = x-term;
    deg (x-term) = 0 <= 0 & x-term in T by Th21,Th24;
    then reconsider r = x-term as Element of T;
    deg r = deg @r = 0 by Th21;
    hence thesis by A3;
  end;
