reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;
reserve X,Y for non empty constituted-DTrees set;

theorem
  C-Subtrees X is empty iff for t being Element of X holds t is root &
  not t.{} in C
proof
  hereby
    assume
A1: C-Subtrees X is empty;
    let t be Element of X;
    reconsider e = {} as Node of t by TREES_1:22;
A2: not t|e in C-Subtrees X by A1;
    then e in Leaves dom t;
    hence t is root & not t.{} in C by A2,Th4;
  end;
  assume
A3: for t being Element of X holds t is root & not t.{} in C;
  assume C-Subtrees X is not empty;
  then reconsider S = C-Subtrees X as non empty Subset of Subtrees X;
  set s = the Element of S;
  consider t being Element of X, n being Node of t such that
  s = t|n and
A4: not n in Leaves dom t or t.n in C by Th24;
  reconsider e = {} as Node of t by TREES_1:22;
  t is root by A3;
  then
A5: dom t = {{}} by TREES_1:29;
  then n = {} by TARSKI:def 1;
  then e^<*0*> in dom t by A3,A4,TREES_1:54;
  hence contradiction by A5,TARSKI:def 1;
end;
