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
  for D1,D2 being non empty set, T being DecoratedTree of D1,D2 holds
  <:T`1,T`2:> = T
proof
  let D1,D2 be non empty set, T be DecoratedTree of D1,D2;
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;
  then
A6: dom <:T`1,T`2:> = dom T by A4,FUNCT_3:50;
  now
    let x be object;
    assume x in dom T;
    then reconsider t = x as Element of dom T;
    thus <:T`1,T`2:>.x = [T`1.t,T`2.t] by A4,A5,FUNCT_3:49
      .= [(T.t)`1,T`2.t] by Th39
      .= [(T.t)`1,(T.t)`2] by Th39
      .= T.x by MCART_1:21;
  end;
  hence thesis by A6;
end;
