
theorem Th9:
  for G being with_terminals non empty DTConstrStr,
  tsg being Element of TS G, s being Terminal of G
  st tsg.{} = s holds tsg = root-tree s
proof
  let G be with_terminals non empty DTConstrStr, tsg be Element of TS G,
  s be Terminal of G;
  assume
A1: tsg.{} = s;
  defpred P[DecoratedTree of the carrier of G] means
  for s being Terminal of G st $1.{} = s holds $1 = root-tree s;
A2: for s being Symbol of G st s in Terminals G holds P[root-tree s]
  by TREES_4:3;
A3: now
    let nt be Symbol of G, ts be FinSequence of TS G;
    assume that
A4: nt ==> roots ts
    and for t being DecoratedTree of the carrier of G st t in rng ts holds P
    [t];
    thus P[nt-tree ts]
    proof
      let s be Terminal of G;
      assume
A5:   (nt-tree ts).{} = s;
A6:   (nt-tree ts).{} = nt by TREES_4:def 4;
A7:   s in Terminals G;
      Terminals G = { t where t is Symbol of G : not ex tnt being FinSequence
      st t ==> tnt } by LANG1:def 2;
      then ex t being Symbol of G st s = t &
      not ex tnt being FinSequence st t ==> tnt by A7;
      hence thesis by A4,A5,A6;
    end;
  end;
  for t being DecoratedTree of the carrier of G st t in TS G holds P[t]
  from DTConstrInd (A2,A3);
  hence thesis by A1;
end;
