reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;
reserve T,T1,T2 for DecoratedTree;

theorem Th33:
  (for x st x in D holds x is Tree) implies union D is Tree
proof
  assume
A1: for x st x in D holds x is Tree;
  then reconsider x = the Element of D as Tree;

 x c= union D by ZFMISC_1:74;
  then reconsider T = union D as non empty set;
 T is Tree-like
  proof
 for X st X in D holds X c= NAT*
    proof
      let X;
      assume X in D;
then    X is Tree by A1;
      hence thesis by TREES_1:def 3;
    end;
    hence T c= NAT* by ZFMISC_1:76;
    thus for p st p in T holds ProperPrefixes p c= T
    proof
      let p;
      assume p in T;
      then consider X such that
A2:   p in X and
A3:   X in D by TARSKI:def 4;
      reconsider X as Tree by A1,A3;
  ProperPrefixes p c= X & X c= T by A2,A3,TREES_1:def 3,ZFMISC_1:74;
      hence thesis;
    end;
    let p,k,n;
    assume that
A4: p^<*k*> in T and
A5: n <= k;
    consider X such that
A6: p^<*k*> in X and
A7: X in D by A4,TARSKI:def 4;
    reconsider X as Tree by A1,A7;
 p^<*n*> in X by A5,A6,TREES_1:def 3;
    hence thesis by A7,TARSKI:def 4;
  end;
  hence thesis;
end;
