reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;
reserve fi,psi for Ordinal-Sequence;

theorem Th44:
  exp(A,succ B) = A*^exp(A,B)
proof
  deffunc D(Ordinal,Ordinal-Sequence) = lim $2;
  deffunc F(Ordinal) = exp(A,$1);
  deffunc C(Ordinal,Ordinal) = A *^ $2;
A1: for B,C holds C = F(B) iff ex fi st C = last fi & dom fi = succ B & fi.
  0 = 1 & (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 Def16;
  for B holds F(succ B) = C(B,F(B)) from OSResultS(A1);
  hence thesis;
end;
