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 Th44:
  T deg<= 0 = Union FreeGen T
  proof
A0: T deg<= 0 = the set of all x-term by Th11;
    thus T deg<= 0 c= Union FreeGen T
    proof
      let x be object; assume x in T deg<= 0;
      then consider s being SortSymbol of S, y being Element of X.s such that
A1:   x = y-term by A0;
      x in FreeGen(s,X) by A1,MSAFREE:def 15;
      then x in (FreeGen T).s & s in the carrier of S = dom FreeGen T
      by PARTFUN1:def 2,MSAFREE:def 16;
      hence thesis by CARD_5:2;
    end;
    let a being object; assume a in Union FreeGen T;
    then consider b such that
A2: b in dom FreeGen T & a in (FreeGen T).b by CARD_5:2;
    reconsider b as SortSymbol of S by A2;
    a in FreeGen(b,X) by A2,MSAFREE:def 16;
    then consider y being set such that
A3: y in X.b & a = root-tree [y,b] by MSAFREE:def 15;
    reconsider y as Element of X.b by A3;
    a = y-term by A3;
    hence a in T deg<= 0 by A0;
  end;
