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 = union the set of all C-Subtrees t where t is Element of X
proof
  hereby
    let x be object;
    assume x in C-Subtrees X;
    then consider t being Element of X such that
A1: ex n being Node of t st x = t|n & (not n in Leaves dom t or t.n in C);
    C-Subtrees t in the set of all C-Subtrees s where s is Element of X
 & x in C-Subtrees t by A1;
    hence x in union the set of all C-Subtrees s where s is Element of X
    by TARSKI:def 4;
  end;
  let x be object;
  assume x in union the set of all C-Subtrees s where s is Element of X;
  then consider Y being set such that
A2: x in Y and
A3: Y in the set of all C-Subtrees s where s is Element of X by
TARSKI:def 4;
  consider t being Element of X such that
A4: Y = C-Subtrees t by A3;
  ex p being Node of t st x = t|p & (not p in Leaves dom t or t.p in C) by A2
,A4;
  hence thesis;
end;
