
theorem
  for nt being Symbol of PeanoNat holds TerminalLanguage nt = {<*0*>}
proof
  let nt be Symbol of PeanoNat;
A1: nt = S or nt = O by Lm2,TARSKI:def 2;
  hereby
    let x be object;
    assume x in TerminalLanguage nt;
    then ex tsg being Element of FinTrees the carrier of PN st
    x = TerminalString tsg & tsg in TS PN & tsg.{} = nt;
    then x = <*O*> by Th13;
    hence x in {<*0*>} by TARSKI:def 1;
  end;
  let x be object;
  assume x in {<*0*>};
  then
A2: x = <*O*> by TARSKI:def 1;
  reconsider prtO = root-tree O as Element of (TS PN) qua non empty set;
  reconsider rtO = root-tree O as Element of TS PN;
  reconsider srtO = <*prtO*> as FinSequence of TS PN;
A3: rtO.{} = O by TREES_4:3;
  then S ==> roots <*rtO*> by Lm5,Th4;
  then
A4: S-tree <*prtO*> in TS PN by Def1;
  then
A5: TerminalString (S-tree srtO) = x by A2,Th13;
A6: (S-tree <*rtO*>).{} = S by TREES_4:def 4;
  TerminalString rtO = x by A2,Th13;
  hence thesis by A1,A3,A4,A5,A6;
end;
