reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th29:
  A <> {} & A is limit_ordinal implies B+^A is limit_ordinal
proof
  assume that
A1: A <> {} and
A2: A is limit_ordinal;
  {} c= A;
  then
A3: {} c< A by A1;
  deffunc F(Ordinal) = B +^ $1;
  consider L being Ordinal-Sequence such that
A4: dom L = A & for C st C in A holds L.C = F(C) from ORDINAL2:sch 3;
A5: B+^A = sup L by A1,A2,A4,ORDINAL2:29
    .= sup rng L;
  for C st C in B+^A holds succ C in B+^A
  proof
    let C such that
A6: C in B+^A;
A7: now
      assume not C in B;
      then consider D such that
A8:   C = B+^D by Th27,ORDINAL1:16;
      now
        assume not D in A;
        then B+^A c= B+^D by ORDINAL1:16,ORDINAL2:33;
        then C in C by A6,A8;
        hence contradiction;
      end;
      then
A9:   succ D in A by A2,ORDINAL1:28;
      then L.(succ D) = B+^succ D by A4
        .= succ(B+^D) by ORDINAL2:28;
      then succ C in rng L by A4,A8,A9,FUNCT_1:def 3;
      hence thesis by A5,ORDINAL2:19;
    end;
    now
      assume C in B;
      then
A10:  succ C c= B by ORDINAL1:21;
A11:  (succ C)+^{} = succ C by ORDINAL2:27;
      B+^{} in B+^A by A3,ORDINAL1:11,ORDINAL2:32;
      hence thesis by A10,A11,ORDINAL1:12,ORDINAL2:34;
    end;
    hence thesis by A7;
  end;
  hence thesis by ORDINAL1:28;
end;
