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 Th29:
  B <> 0 & B is limit_ordinal implies for fi st dom fi = B & for
  C st C in B holds fi.C = A+^C holds A+^B = sup fi
proof
  deffunc C(Ordinal,Ordinal) = succ $2;
  deffunc D(Ordinal,Ordinal-Sequence) = sup $2;
  assume
A1: B <> 0 & B is limit_ordinal;
  deffunc F(Ordinal) = A+^$1;
  let fi 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 st C = last fi & dom fi = succ B & fi.
  0 = A & (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 Def14;
  thus F(B) = D(B,fi) from OSResultL(A4,A1,A2,A3);
end;
