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 Th6:
  dom fi = dom psi & B is_limes_of fi & C is_limes_of psi & ((for A
st A in dom fi holds fi.A c= psi.A) or for A st A in dom fi holds fi.A in psi.A
  ) implies B c= C
proof
  assume that
A1: dom fi = dom psi and
A2: B = 0 & (ex A1 st A1 in dom fi & for C st A1 c= C & C in dom fi
holds fi.C = 0) or B <> 0 & for A1,C st A1 in B & B in C ex D st D in dom fi
  & for E being Ordinal st D c= E & E in dom fi holds A1 in fi.E & fi.E in C
  and
A3: C = 0 & (ex B st B in dom psi & for A1 st B c= A1 & A1 in dom psi
  holds psi.A1 = 0) or C <> 0 & for B,A1 st B in C & C in A1 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
  A1 and
A4: (for A st A in dom fi holds fi.A c= psi.A) or for A st A in dom fi
  holds fi.A in psi.A;
A5: for A st A in dom fi holds fi.A c= psi.A by A4,ORDINAL1:def 2;
  now
    per cases;
    suppose
      B = {} & C = {};
      hence thesis;
    end;
    suppose
      B = {} & C <> {};
      then B in C by ORDINAL3:8;
      hence thesis by ORDINAL1:def 2;
    end;
    suppose
A6:   B <> {} & C = {};
      then {} in B by ORDINAL3:8;
      then consider A2 being Ordinal such that
A7:   A2 in dom fi and
A8:   for A st A2 c= A & A in dom fi holds {} in fi.A & fi.A in succ
      B by A2,ORDINAL1:6;
      consider A1 such that
A9:   A1 in dom psi and
A10:  for A st A1 c= A & A in dom psi holds psi.A = {} by A3,A6;
A11:  A1 \/ A2 = A1 or A1 \/ A2 = A2 by ORDINAL3:12;
      then
A12:  fi.(A1 \/ A2) c= psi.(A1 \/ A2) by A1,A5,A9,A7;
      A2 c= A1 \/ A2 by XBOOLE_1:7;
      then {} in fi.(A1 \/ A2) by A1,A9,A7,A8,A11;
      hence thesis by A1,A9,A10,A7,A11,A12,XBOOLE_1:7;
    end;
    suppose
A13:  B <> {} & C <> {};
      assume not B c= C;
      then C in B by ORDINAL1:16;
      then consider A1 such that
A14:  A1 in dom fi and
A15:  for A st A1 c= A & A in dom fi holds C in fi.A & fi.A in succ B
      by A2,ORDINAL1:6;
      {} in C by A13,ORDINAL3:8;
      then consider A2 being Ordinal such that
A16:  A2 in dom psi and
A17:  for A st A2 c= A & A in dom psi holds {} in psi.A & psi.A in
      succ C by A3,ORDINAL1:6;
A18:  A1 \/ A2 = A1 or A1 \/ A2 = A2 by ORDINAL3:12;
      reconsider A3 = psi.(A1 \/ A2) as Ordinal;
      A2 c= A1 \/ A2 by XBOOLE_1:7;
      then psi.(A1 \/ A2) in succ C by A1,A14,A16,A17,A18;
      then
A19:  A3 c= C by ORDINAL1:22;
      A1 c= A1 \/ A2 by XBOOLE_1:7;
      then
A20:  C in fi.(A1 \/ A2) by A1,A14,A15,A16,A18;
      fi.(A1 \/ A2) c= psi.(A1 \/ A2) by A1,A5,A14,A16,A18;
      hence contradiction by A20,A19,ORDINAL1:5;
    end;
  end;
  hence thesis;
end;
