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 Th30:
  1 in a implies a |^|^ omega is limit_ordinal proof assume
A1: 1 in a;
    deffunc F(Ordinal) = a |^|^ $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;
    phi is increasing
    proof
      let b,c; assume
A3:   b in c & c in dom phi; then
      reconsider b,c as Element of NAT by A2,ORDINAL1:10;
      b in Segm c by A3;
      then b < c by NAT_1:44; then
      phi.b = F(b) & F(b) in F(c) by A1,A2,Th24;
      hence thesis by A2;
    end; then
    lim phi = sup phi & sup phi is limit_ordinal by A2,ORDINAL4:8,16;
    hence thesis by A2,Th15;
  end;
