
theorem Th16:
  for G being non empty DTConstrStr for t being set st t in TS G holds
  (ex d being Symbol of G st d in Terminals G & t = root-tree d) or
  ex o being Symbol of G, p being FinSequence of TS G
  st o ==> roots p & t = o-tree p
proof
  let G be non empty DTConstrStr;
  let t be set;
  assume that
A1: t in TS G and
A2: not ex d being Symbol of G st d in Terminals G & t = root-tree d and
A3: not ex o being Symbol of G, p being FinSequence of TS G
  st o ==> roots p & t = o-tree p;
A4: (TS G) \ {t} c= TS G by XBOOLE_1:36;
  reconsider Y = (TS G) \ {t} as Subset of FinTrees the carrier of G;
A5: now
    let d be Symbol of G;
    assume
A6: d in Terminals G;
    then
A7: root-tree d in TS G by DTCONSTR:def 1;
    root-tree d <> t by A2,A6;
    hence root-tree d in Y by A7,ZFMISC_1:56;
  end;
  now
    let o be Symbol of G, p be FinSequence of Y;
    rng p c= Y by FINSEQ_1:def 4;
    then rng p c= TS G by A4;
    then reconsider q = p as FinSequence of TS G by FINSEQ_1:def 4;
    assume
A8: o ==> roots p;
    then
A9: o-tree q in TS G by DTCONSTR:def 1;
    t <> o-tree q by A3,A8;
    hence o-tree p in Y by A9,ZFMISC_1:56;
  end;
  then TS G c= Y by A5,DTCONSTR:def 1;
  then t nin {t} by A1,XBOOLE_0:def 5;
  hence contradiction by TARSKI:def 1;
end;
