
theorem Th13:
  for t being DecoratedTree of the carrier of PeanoNat
  st t in TS PeanoNat holds TerminalString t = <*0*>
proof
  consider f being Function of (TS PN), (Terminals PN)* such that
A1: TerminalString root-tree (O qua Symbol of PN) = f.(root-tree O) and
A2: for t being Symbol of PN st t in Terminals PN holds f.(root-tree t)
  = <*t*> and
A3: for nt being Symbol of PN, ts being FinSequence of TS(PN) st nt ==>
  roots ts holds f.(nt-tree ts) = FlattenSeq(f * ts)
  by Def11;
  defpred P[DecoratedTree of the carrier of PN] means
  TerminalString $1 = <*0*>;
A4: now
    let s be Symbol of PN;
    assume s in Terminals PN;
    then s = O by Lm10,TARSKI:def 1;
    hence P[root-tree s] by A1,A2;
  end;
A5: now
    let nt be Symbol of PN, ts be FinSequence of TS PN;
    assume that
A6: nt ==> roots ts and
A7: for t being DecoratedTree of the carrier of PN st t in rng ts holds P[t];
A8: nt-tree ts in TS PN by A6,Def1;
    roots ts = <*O*> or roots ts = <*1*> by A6,Def2;
    then len roots ts = 1 by FINSEQ_1:40;
    then consider x being Element of FinTrees the carrier of PN such that
A9: ts = <*x*> and
A10: x in TS PN by Th5;
    reconsider x9 = x as Element of TS PN by A10;
    rng ts = {x} by A9,FINSEQ_1:39;
    then
A11: x in rng ts by TARSKI:def 1;
    f.x9 is Element of (Terminals PN)*;
    then
A12: f.x9 = TerminalString x by A2,A3,Def11
      .= <*O*> by A7,A11;
    f * ts = <*f.x*> by A9,FINSEQ_2:35;
    then f.(nt-tree ts) = FlattenSeq(<*f.x9*>) by A3,A6
      .= <*O*> by A12,PRE_POLY:1;
    hence P[nt-tree ts] by A2,A3,A8,Def11;
  end;
  thus for t being DecoratedTree of the carrier of PN
  st t in TS PN holds P[t] from DTConstrInd(A4,A5);
end;
