reserve x for set,
  t,t1,t2 for DecoratedTree;
reserve C for set;

theorem
  C-Subtrees t is empty iff t is root & not t.{} in C
proof
  reconsider e = {} as Node of t by TREES_1:22;
  hereby
    assume C-Subtrees t is empty;
    then
A1: not t|e in C-Subtrees t;
    then e in Leaves dom t;
    hence t is root & not t.{} in C by A1,Th4;
  end;
  assume that
A2: t is root and
A3: not t.{} in C;
  assume C-Subtrees t is not empty;
  then reconsider S = C-Subtrees t as non empty Subset of Subtrees t;
  set s = the Element of S;
  consider n being Node of t such that
  s = t|n and
A4: not n in Leaves dom t or t.n in C by Th17;
A5: dom t = {{}} by A2,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;
