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 Th43:
  b <> 0 & b is limit_ordinal implies
  for phi being Ordinal-Sequence st dom phi = b &
     for c st c in b holds phi.c = epsilon_c
  holds epsilon_b = lim phi
  proof
    assume
A1: b <> 0 & b is limit_ordinal;
    deffunc F(Ordinal) = epsilon_$1;
    deffunc D(Ordinal,Ordinal-Sequence) = lim $2;
    deffunc C(Ordinal,Ordinal) = $2|^|^omega;
    let fi being Ordinal-Sequence such that
A2: dom fi = b and
A3: for c st c in b holds fi.c = F(c);
A4: 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;
    thus F(b) = D(b,fi) from ORDINAL2:sch 16(A4,A1,A2,A3);
  end;
