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
  for i holds T deg<= i c= Free(S,X) deg<= i
  proof
    defpred P[Nat] means T deg<= $1 c= Free(S,X) deg<= $1;
    T deg<= 0 = Union FreeGen T & Free(S,X) deg<= 0 = Union FreeGen Free(S,X)
    by Th44;
    then
A0: P[0];
A1: now let i; assume P[i];
      thus P[i+1]
      proof
        let x be object; assume x in T deg<= (i+1);
        then consider r such that
A3:     x = r & deg r <= i+1;
        reconsider t = r as Element of Free(S,X) by MSAFREE4:39;
        deg t = deg r;
        hence thesis by A3;
      end;
    end;
    thus for i holds P[i] from NAT_1:sch 2(A0,A1);
  end;
