
theorem
  for T1, T2 being DecoratedTree holds roots <*T1, T2*> = <*T1.{}, T2.{}*>
proof
  let T1, T2 be DecoratedTree;
A1: len <*T1, T2*> = 2 by FINSEQ_1:44;
A2: len <*T1.{}, T2.{}*> = 2 by FINSEQ_1:44;
A7: dom <*T1, T2*> = Seg 2 by A1,FINSEQ_1:def 3;
A8: dom <*T1.{}, T2.{}*> = Seg 2 by A2,FINSEQ_1:def 3;
  now
    let i be Element of NAT;
    assume i in dom <*T1, T2*>;
    then i in Seg 2 by A1,FINSEQ_1:def 3;
    then i = 1 or i = 2 by FINSEQ_1:2,TARSKI:def 2;
    hence ex t being DecoratedTree st
    t = <*T1, T2*>.i & <*T1.{}, T2.{}*>.i = t.{};
  end;
  hence thesis by A7,A8,TREES_3:def 18;
end;
