reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;
reserve T,T9 for DecoratedTree,
  x,y for set;
reserve D1, D2 for non empty set,
  T for DecoratedTree of D1,D2,
  d1 for Element of D1,
  d2 for Element of D2,
  F for non empty DTree-set of D1,D2,
  F1 for non empty (DTree-set of D1),
  F2 for non empty DTree-set of D2;

theorem Th24:
  for D1, D2, T holds dom T`1 = dom T & dom T`2 = dom T
proof
  let D1, D2, T;
  A1: T`1 = pr1(D1,D2)*T & T`2 = pr2(D1,D2)*T by TREES_3:def 12,def 13;
  A2: rng T c= [:D1,D2:] & dom pr1(D1,D2) = [:D1,D2:] by FUNCT_2:def 1
,RELAT_1:def 19;
 dom pr2(D1,D2) = [:D1,D2:] by FUNCT_2:def 1;
  hence thesis by A1,A2,RELAT_1:27;
end;
