
theorem
  for pn being Element of TS(PeanoNat) holds pn = NAT-to-PN.(PN-to-NAT.pn)
proof
  defpred P[DecoratedTree of the carrier of PN] means
  $1 = NAT-to-PN.(PN-to-NAT.$1);
A1: now
    let s be Symbol of PN;
    assume
A2: s in Terminals PN;
    then NAT-to-PN.(PN-to-NAT.(root-tree s)) = NAT-to-PN.0 by Def8
      .= root-tree O by Def10;
    hence P[root-tree s] by A2,Lm10,TARSKI:def 1;
  end;
A3: now
    let nt be Symbol of PN, ts be FinSequence of TS(PN);
    assume that
A4: nt ==> roots ts and
A5: for t being DecoratedTree of the carrier of PN st t in rng ts holds P[t];
A6: nt <> O by A4,Lm8;
    roots ts = <*O*> or roots ts = <*S*> by A4,Def2;
    then len roots ts = 1 by FINSEQ_1:40;
    then consider dt being Element of FinTrees the carrier of PN such that
A7: ts = <*dt*> and
A8: dt in TS(PN) by Th5;
    reconsider dt as Element of TS(PN) by A8;
    rng ts = {dt} by A7,FINSEQ_1:38;
    then dt in rng ts by TARSKI:def 1;
    then
A9: dt = NAT-to-PN.(PN-to-NAT.dt) by A5;
A10: PN-to-NAT * ts = <*PN-to-NAT.dt*> by A7,FINSEQ_2:35;
    set N = PN-to-NAT.dt;
A11: plus-one(<*N*>) = N+1;
    NAT-to-PN.(N+1) = PNsucc dt by A9,Def10
      .= nt-tree ts by A6,A7,Lm2,TARSKI:def 2;
    hence P[nt-tree ts] by A4,A10,A11,Def8;
  end;
  for t being DecoratedTree of the carrier of PN
  st t in TS(PN) holds P[t] from DTConstrInd (A1,A3);
  hence thesis;
end;
