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 Th39:
  for e being epsilon Ordinal st 0 in n holds
  e |^|^ (n+2) = exp(omega, e |^|^ (n+1))
  proof
    let e be epsilon Ordinal such that
A1: 0 in n;
    0 in e by ORDINAL3:8; then
    omega in e & e c= e|^|^n by A1,Th23,Th37; then
A2: omega c= e|^|^n by ORDINAL1:def 2;
    thus e |^|^ (n+2) = e|^|^Segm(n+1+1)
    .= e|^|^succ Segm(n+1) by NAT_1:38
    .= exp(e, e|^|^(n+1)) by Th14
    .= exp(exp(omega, e), e|^|^(n+1)) by Def5
    .= exp(omega, (e|^|^Segm(n+1))*^e) by ORDINAL4:31
    .= exp(omega, (e|^|^succ Segm n)*^e) by NAT_1:38
    .= exp(omega, exp(e, e|^|^n)*^e) by Th14
    .= exp(omega, exp(e, e|^|^n)*^exp(e,1)) by ORDINAL2:46
    .= exp(omega, exp(e, 1+^e|^|^n)) by ORDINAL4:30
    .= exp(omega, exp(e, e|^|^n)) by A2,CARD_2:74
    .= exp(omega, e|^|^succ Segm n) by Th14
    .= exp(omega, e |^|^ Segm(n+1)) by NAT_1:38
    .= exp(omega, e |^|^ (n+1));
  end;
