reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem
  for n being Nat st n > 1 holds n |^|^ omega = omega
  proof
    let n be Nat such that
A1: n > 1;
    deffunc F(Ordinal) = n |^|^ $1;
    consider phi being Ordinal-Sequence such that
A2: dom phi = omega & for b st b in omega holds phi.b = F(b)
    from ORDINAL2:sch 3;
A3: rng phi c= omega
    proof
      let x be object; assume x in rng phi; then
      consider a being object such that
A4:   a in dom phi & x = phi.a by FUNCT_1:def 3;
      reconsider a as Element of omega by A2,A4;
      x = n |^|^ a by A2,A4;
      hence thesis by ORDINAL1:def 12;
    end;
A5: n |^|^ omega = lim phi by A2,Th15;
    now
      thus {} <> omega;
      let b,c such that
A6:   b in omega & omega in c;
      reconsider x = b as Element of omega by A6;
      take D = n |^|^ x; thus D in dom phi by A2,ORDINAL1:def 12;
      x < D by A1,Th28; then
A7:   b in Segm D by NAT_1:44;
      let E be Ordinal;
      assume
A8:   D c= E & E in dom phi; then
      reconsider e = E as Element of omega by A2;
      x in Segm e by A7,A8; then
      x < e & e < n |^|^ e by A1,Th28,NAT_1:44; then
A9:   x < n |^|^ e & phi.e = F(e) by A2,XXREAL_0:2;
      phi.E in rng phi by A8,FUNCT_1:def 3;
      then reconsider phiE = phi.E as Nat by A3;
    b in Segm phiE by A9,NAT_1:44;
      hence b in phi.E;
      phi.E in rng phi by A8,FUNCT_1:def 3;
      hence phi.E in c by A6,A3,ORDINAL1:10;
    end; then
    omega is_limes_of phi by ORDINAL2:def 9;
    hence n |^|^ omega = omega by A5,ORDINAL2:def 10;
  end;
