reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;

theorem
  A is_limes_of fi implies B+^A is_limes_of B+^fi
proof
  assume that
A1: A = 0 & (ex B st B in dom fi & for C st B c= C & C in dom fi holds
fi.C = 0) or A <> 0 & for B,C st B in A & A in C ex D st D in dom fi & for E
  being Ordinal st D c= E & E in dom fi holds B in fi.E & fi.E in C;
A2: dom fi = dom(B+^fi) by ORDINAL3:def 1;
  per cases;
  case
A3: B+^A = 0;
    then consider A1 such that
A4: A1 in dom fi and
A5: for C st A1 c= C & C in dom fi holds fi.C = {} by A1,ORDINAL3:26;
    take A1;
    thus A1 in dom(B+^fi) by A4,ORDINAL3:def 1;
    let C;
    assume that
A6: A1 c= C and
A7: C in dom(B+^fi);
A8: (B+^fi).C = B+^(fi.C) by A2,A7,ORDINAL3:def 1;
    fi.C = {} by A2,A5,A6,A7;
    hence thesis by A3,A8,ORDINAL3:26;
  end;
  case
    B+^A <> 0;
    now
      per cases;
      suppose
A9:     A = {};
        then consider A1 such that
A10:    A1 in dom fi and
A11:    for C st A1 c= C & C in dom fi holds fi.C = {} by A1;
        let B1,B2 be Ordinal such that
A12:    B1 in B+^A and
A13:    B+^A in B2;
        take A1;
        thus A1 in dom(B+^fi) by A10,ORDINAL3:def 1;
        let C;
        assume that
A14:    A1 c= C and
A15:    C in dom(B+^fi);
        (B+^fi).C = B+^(fi.C) by A2,A15,ORDINAL3:def 1;
        hence B1 in (B+^fi).C & (B+^fi).C in B2 by A2,A9,A11,A12,A13,A14,A15;
      end;
      suppose
A16:    A <> {};
        let B1,B2 be Ordinal;
        assume that
A17:    B1 in B+^A and
A18:    B+^A in B2;
        B1-^B in A by A16,A17,ORDINAL3:60;
        then consider A1 such that
A19:    A1 in dom fi and
A20:    for C st A1 c= C & C in dom fi holds B1-^B in fi.C & fi.C in
        B2-^B by A1,A18,ORDINAL3:61;
A21:    B1 c= B+^(B1-^B) by ORDINAL3:62;
A22:    B c= B+^A by ORDINAL3:24;
        B+^A c= B2 by A18,ORDINAL1:def 2;
        then B c= B2 by A22;
        then
A23:    B+^(B2-^B) = B2 by ORDINAL3:def 5;
        take A1;
        thus A1 in dom(B+^fi) by A19,ORDINAL3:def 1;
        let C;
        assume that
A24:    A1 c= C and
A25:    C in dom(B+^fi);
A26:    (B+^fi).C = B+^(fi.C) by A2,A25,ORDINAL3:def 1;
        reconsider E = fi.C as Ordinal;
        B1-^B in E by A2,A20,A24,A25;
        then
A27:    B+^(B1-^B) in B+^E by ORDINAL2:32;
        E in B2-^B by A2,A20,A24,A25;
        hence B1 in (B+^fi).C & (B+^fi).C in B2 by A21,A26,A23,A27,ORDINAL1:12
,ORDINAL2:32;
      end;
    end;
    hence thesis;
  end;
end;
