theorem Th12:
  for x being Element of (the carrier of X)*
  ex s st |.s.| = x
  proof
    set D = the carrier of X;
    defpred P[FinSequence of D] means ex s st |.s.| = $1;
A1: P[<*> D]
    proof
      consider s such that
A2:   emp s by Th2;
      take s; thus thesis by A2,Th5;
    end;
A3: for p being FinSequence of D for x being Element of D st P[p]
    holds P[<*x*>^p]
    proof
      let p be FinSequence of D, x be Element of D;
      given s such that
A4:   |.s.| = p;
      take s2 = push(x,s);
      thus thesis by A4,Th8;
    end;
    for p being FinSequence of D holds P[p] from IndSeqD(A1,A3);
    hence thesis;
  end;
