reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;

theorem Th3:
  for P holds P^{{}} = P & {{}}^P = P
proof
  let P;
  A1: for a holds a in P^{{}} iff a in P
    proof
    let a;
    thus a in P^{{}} implies a in P
    proof
      assume a in P^{{}};
      then consider p, q such that
  A2: a = p^q and A3: p in P and A4: q in {{}} by Def2;
      q = {} by A4, TARSKI:def 1;
      hence thesis by A2, A3, FINSEQ_1:34;
    end;
      assume A10: a in P;
      then reconsider a as FinSequence;
      {} in {{}} by TARSKI:def 1;
      then a^{} in P^{{}} by A10, Def2;
      hence thesis by FINSEQ_1:34;
    end;
  for a holds a in {{}}^P iff a in P
    proof
    let a;
    thus a in {{}}^P implies a in P
      proof
      assume a in {{}}^P;
      then consider q, p such that
 A12: a = q^p and A13: q in {{}} and A14: p in P by Def2;
      q = {} by A13, TARSKI:def 1;
      hence thesis by A12, A14, FINSEQ_1:34;
      end;
      assume A20: a in P;
      then reconsider a as FinSequence;
      {} in {{}} by TARSKI:def 1;
      then {}^a in {{}}^P by A20, Def2;
      hence thesis by FINSEQ_1:34;
    end;
  hence thesis by A1, TARSKI:2;
end;
