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 Th42:
  epsilon_(succ a) = epsilon_a |^|^ omega
  proof
    deffunc F(Ordinal) = epsilon_$1;
    deffunc D(Ordinal,Ordinal-Sequence) = lim $2;
    deffunc C(Ordinal,Ordinal) = $2|^|^omega;
A1: for b,c holds c = F(b) iff
    ex fi being Ordinal-Sequence st c = last fi & dom fi = succ b &
    fi.0 = omega|^|^omega &
    (for c st succ c in succ b holds fi.succ c = C(c,fi.c)) &
    for c st c in succ b & c <> 0 & c is limit_ordinal
    holds fi.c = D(c,fi|c) by Def7;
    for b holds F(succ b) = C(b,F(b)) from ORDINAL2:sch 15(A1);
    hence thesis;
  end;
