reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th39:
  for D1,D2 being non empty set, T being DecoratedTree of D1,D2,
  t being Element of dom T holds (T.t)`1 = T`1.t & T`2.t = (T.t)`2
proof
  let D1,D2 be non empty set, T be DecoratedTree of D1,D2;
  let t be Element of dom T;
A1: dom pr1(D1,D2) = [:D1,D2:] by FUNCT_2:def 1;
A2: dom pr2(D1,D2) = [:D1,D2:] by FUNCT_2:def 1;
A3: rng T c= [:D1,D2:];
  then
A4: dom T`1 = dom T by A1,RELAT_1:27;
A5: dom T`2 = dom T by A2,A3,RELAT_1:27;
A6: T.t = [(T.t)`1,(T.t)`2] by MCART_1:21;
  then
A7: T`1.t = pr1(D1,D2).((T.t)`1,(T.t)`2) by A4,FUNCT_1:12;
  T`2.t = pr2(D1,D2).((T.t)`1,(T.t)`2) by A5,A6,FUNCT_1:12;
  hence thesis by A7,FUNCT_3:def 4,def 5;
end;
