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 Th5:
  A is_limes_of fi implies A*^B is_limes_of fi*^B
proof
A1: dom fi = dom(fi*^B) by ORDINAL3:def 4;
  assume
A2: A is_limes_of fi;
  then
A3: dom fi <> {} by Lm4;
  per cases;
  case
    A*^B = 0;
    then
A4: B = {} or A = {} by ORDINAL3:31;
    now
      per cases;
      suppose
A5:     B = {};
        set x = the Element of dom fi;
        reconsider x as Ordinal;
        take A1 = x;
        thus A1 in dom(fi*^B) by A1,A3;
        let C;
        assume that
        A1 c= C and
A6:     C in dom(fi*^B);
        thus (fi*^B).C = (fi.C)*^B by A1,A6,ORDINAL3:def 4
          .= {} by A5,ORDINAL2:38;
      end;
      suppose
        B <> {};
        then consider A1 such that
A7:     A1 in dom fi and
A8:     for C st A1 c= C & C in dom fi holds fi.C = {} by A2,A4,ORDINAL2:def 9;
        take A1;
        thus A1 in dom(fi*^B) by A7,ORDINAL3:def 4;
        let C;
        assume that
A9:     A1 c= C and
A10:    C in dom(fi*^B);
A11:    (fi*^B).C = (fi.C)*^B by A1,A10,ORDINAL3:def 4;
        fi.C = {} by A1,A8,A9,A10;
        hence (fi*^B).C = {} by A11,ORDINAL2:35;
      end;
    end;
    hence thesis;
  end;
  case
A12: A*^B <> 0;
    let B1,B2 be Ordinal such that
A13: B1 in A*^B and
A14: A*^B in B2;
A15: B <> {} by A12,ORDINAL2:38;
A16: now
      assume not ex A1 st A = succ A1;
      then A is limit_ordinal by ORDINAL1:29;
      then consider C such that
A17:  C in A and
A18:  B1 in C*^B by A13,ORDINAL3:41;
      A in succ A by ORDINAL1:6;
      then consider D such that
A19:  D in dom fi and
A20:  for A1 st D c= A1 & A1 in dom fi holds C in fi.A1 & fi.A1 in
      succ A by A2,A17,ORDINAL2:def 9;
      take D;
      thus D in dom(fi*^B) by A19,ORDINAL3:def 4;
      let A1;
      assume that
A21:  D c= A1 and
A22:  A1 in dom(fi*^B);
      reconsider E = fi.A1 as Ordinal;
      fi.A1 in succ A by A1,A20,A21,A22;
      then
A23:  E c= A by ORDINAL1:22;
      C in fi.A1 by A1,A20,A21,A22;
      then C*^B in E*^B by A15,ORDINAL2:40;
      then
A24:  B1 in E*^B by A18,ORDINAL1:10;
      (fi*^B).A1 = (fi.A1)*^B by A1,A22,ORDINAL3:def 4;
      hence B1 in (fi*^B).A1 & (fi*^B).A1 in B2 by A14,A23,A24,ORDINAL1:12
,ORDINAL2:41;
    end;
    now
A25:  A in succ A by ORDINAL1:6;
      given A1 such that
A26:  A = succ A1;
      A1 in A by A26,ORDINAL1:6;
      then consider D such that
A27:  D in dom fi and
A28:  for C st D c= C & C in dom fi holds A1 in fi.C & fi.C in succ A
      by A2,A25,ORDINAL2:def 9;
      take D;
      thus D in dom(fi*^B) by A27,ORDINAL3:def 4;
      let C;
      assume that
A29:  D c= C and
A30:  C in dom(fi*^B);
      reconsider E = fi.C as Ordinal;
      A1 in E by A1,A28,A29,A30;
      then
A31:  A c= E by A26,ORDINAL1:21;
      E in succ A by A1,A28,A29,A30;
      then
A32:  E c= A by ORDINAL1:22;
      (fi*^B).C = E*^B by A1,A30,ORDINAL3:def 4;
      hence B1 in (fi*^B).C & (fi*^B).C in B2 by A13,A14,A31,A32,
XBOOLE_0:def 10;
    end;
    hence thesis by A16;
  end;
end;
