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;
reserve f1,f2 for Ordinal-Sequence;

theorem
  dom f1 = dom fi & dom fi = dom f2 & A is_limes_of f1 & A is_limes_of
  f2 & (for A st A in dom fi holds f1.A c= fi.A & fi.A c= f2.A) implies A
  is_limes_of fi
proof
  assume that
A1: dom f1 = dom fi and
A2: dom fi = dom f2 and
A3: A = 0 & (ex B st B in dom f1 & for C st B c= C & C in dom f1 holds
f1.C = 0) or A <> 0 & for B,C st B in A & A in C ex D st D in dom f1 & for E
  being Ordinal st D c= E & E in dom f1 holds B in f1.E & f1.E in C and
A4: A = 0 & (ex B st B in dom f2 & for C st B c= C & C in dom f2 holds
f2.C = 0) or A <> 0 & for B,C st B in A & A in C ex D st D in dom f2 & for E
  being Ordinal st D c= E & E in dom f2 holds B in f2.E & f2.E in C and
A5: for A st A in dom fi holds f1.A c= fi.A & fi.A c= f2.A;
  per cases;
  case
    A = 0;
    then consider B being Ordinal such that
A6: B in dom f2 and
A7: for C st B c= C & C in dom f2 holds f2.C = {} by A4;
    take B;
    thus B in dom fi by A2,A6;
    let C;
    assume that
A8: B c= C and
A9: C in dom fi;
    f2.C = {} by A2,A7,A8,A9;
    hence thesis by A5,A9,XBOOLE_1:3;
  end;
  case
    A <> 0;
    let B,C;
    assume that
A10: B in A and
A11: A in C;
    consider D2 being Ordinal such that
A12: D2 in dom f2 and
A13: for A1 st D2 c= A1 & A1 in dom f2 holds B in f2.A1 & f2.A1 in C
    by A4,A10,A11;
    consider D1 being Ordinal such that
A14: D1 in dom f1 and
A15: for A1 st D1 c= A1 & A1 in dom f1 holds B in f1.A1 & f1.A1 in C
    by A3,A10,A11;
    take D = D1 \/ D2;
    thus D in dom fi by A1,A2,A14,A12,ORDINAL3:12;
    let A1;
    assume that
A16: D c= A1 and
A17: A1 in dom fi;
    reconsider B1 = fi.A1, B2 = f2.A1 as Ordinal;
A18: B1 c= B2 by A5,A17;
    D2 c= D by XBOOLE_1:7;
    then D2 c= A1 by A16;
    then
A19: B2 in C by A2,A13,A17;
    D1 c= D by XBOOLE_1:7;
    then D1 c= A1 by A16;
    then
A20: B in f1.A1 by A1,A15,A17;
    f1.A1 c= fi.A1 by A5,A17;
    hence thesis by A20,A18,A19,ORDINAL1:12;
  end;
end;
