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 Th3:
  A is_limes_of psi implies A is_limes_of fi^psi
proof
  assume that
A1: A = 0 & (ex B st B in dom psi & for C st B c= C & C in dom psi
holds psi.C = 0) or A <> 0 & for B,C st B in A & A in C ex D st D in dom psi
  & for E being Ordinal st D c= E & E in dom psi holds B in psi.E & psi.E in C;
A2: dom(fi^psi) = (dom fi)+^(dom psi) by Def1;
  per cases;
  case
    A = 0;
    then consider B such that
A3: B in dom psi and
A4: for C st B c= C & C in dom psi holds psi.C = {} by A1;
    take B1 = (dom fi)+^B;
    thus B1 in dom(fi^psi) by A2,A3,ORDINAL2:32;
    let C;
    assume that
A5: B1 c= C and
A6: C in dom(fi^psi);
A7: C = B1+^(C-^B1) by A5,ORDINAL3:def 5
      .= (dom fi)+^(B+^(C-^B1)) by ORDINAL3:30;
    then
A8: B+^(C-^B1) in dom psi by A2,A6,ORDINAL3:22;
    B c= B+^(C-^B1) by ORDINAL3:24;
    then psi.(B+^(C-^B1)) = {} by A2,A4,A6,A7,ORDINAL3:22;
    hence thesis by A7,A8,Def1;
  end;
  case
    A <> 0;
    let B,C;
    assume that
A9: B in A and
A10: A in C;
    consider D such that
A11: D in dom psi and
A12: for E being Ordinal st D c= E & E in dom psi holds B in psi.E &
    psi.E in C by A1,A9,A10;
    take D1 = (dom fi)+^D;
    thus D1 in dom(fi^psi) by A2,A11,ORDINAL2:32;
    let E be Ordinal;
    assume that
A13: D1 c= E and
A14: E in dom(fi^psi);
A15: D c= D+^(E-^D1) by ORDINAL3:24;
A16: E = D1+^(E-^D1) by A13,ORDINAL3:def 5
      .= (dom fi)+^(D+^(E-^D1)) by ORDINAL3:30;
    then
A17: D+^(E-^D1) in dom psi by A2,A14,ORDINAL3:22;
    then (fi^psi).E = psi.(D+^(E-^D1)) by A16,Def1;
    hence thesis by A12,A15,A17;
  end;
end;
