
theorem Th4:
  for T being DecoratedTree holds roots <*T*> = <*T.{}*>
proof
  let T be DecoratedTree;
A1: dom <*T*> = Seg 1 by FINSEQ_1:def 8;
A2: dom <*T.{}*> = Seg 1 by FINSEQ_1:def 8;
A3: <*T*>.1 = T;
A4: <*T.{}*>.1 = T.{};
  now
    let i be Element of NAT;
    assume
A5: i in dom <*T*>;
    take t = T;
    thus t = <*T*>.i & <*T.{}*>.i = t.{} by A1,A3,A4,A5,FINSEQ_1:2,TARSKI:def 1
;
  end;
  hence thesis by A1,A2,TREES_3:def 18;
end;
