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 Th48:
  a c= epsilon_a
  proof
    defpred P[Ordinal] means $1 c= epsilon_$1;
A1: P[0];
A2: P[b] implies P[succ b] proof assume
A3:   P[b];
      epsilon_b in epsilon_succ b by Th44,ORDINAL1:6; then
      b in epsilon_succ b by A3,ORDINAL1:12;
      hence thesis by ORDINAL1:21;
    end;
A4: c <> 0 & c is limit_ordinal & (for b st b in c holds P[b]) implies P[c]
    proof assume that
      c <> 0 & c is limit_ordinal and
A5:   for b st b in c holds P[b];
      let x be Ordinal; assume
A6:   x in c;
      reconsider a = x as Ordinal;
      P[a] & epsilon_a in epsilon_c by A5,A6,Th44;
      hence thesis by ORDINAL1:12;
    end;
    P[b] from ORDINAL2:sch 1(A1,A2,A4);
    hence thesis;
  end;
