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 Th14:
  f1 is increasing & A is_limes_of f2 & sup rng f1 = dom f2 & fi =
  f2*f1 implies A is_limes_of fi
proof
  assume that
A1: f1 is increasing and
A2: 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
A3: sup rng f1 = dom f2 and
A4: fi = f2*f1;
  per cases;
  case
    A = 0;
    then consider B such that
A5: B in dom f2 and
A6: for C st B c= C & C in dom f2 holds f2.C = {} by A2;
    consider B1 being Ordinal such that
A7: B1 in rng f1 and
A8: B c= B1 by A3,A5,ORDINAL2:21;
    consider x being object such that
A9: x in dom f1 and
A10: B1 = f1.x by A7,FUNCT_1:def 3;
    reconsider x as Ordinal by A9;
    take x;
    B1 in dom f2 by A3,A7,ORDINAL2:19;
    hence x in dom fi by A4,A9,A10,FUNCT_1:11;
    let C such that
A11: x c= C and
A12: C in dom fi;
    reconsider C1 = f1.C as Ordinal;
A13: dom fi c= dom f1 by A4,RELAT_1:25;
    then B1 c= C1 by A1,A10,A11,A12,Th9;
    then
A14: B c= C1 by A8;
    C1 in rng f1 by A12,A13,FUNCT_1:def 3;
    then f2.C1 = {} by A3,A6,A14,ORDINAL2:19;
    hence thesis by A4,A12,FUNCT_1:12;
  end;
  case
    A <> 0;
    let B,C;
    assume that
A15: B in A and
A16: A in C;
    consider D such that
A17: D in dom f2 and
A18: for A1 st D c= A1 & A1 in dom f2 holds B in f2.A1 & f2.A1 in C by A2,A15
,A16;
    consider B1 being Ordinal such that
A19: B1 in rng f1 and
A20: D c= B1 by A3,A17,ORDINAL2:21;
    consider x being object such that
A21: x in dom f1 and
A22: B1 = f1.x by A19,FUNCT_1:def 3;
    reconsider x as Ordinal by A21;
    take x;
    B1 in dom f2 by A3,A19,ORDINAL2:19;
    hence x in dom fi by A4,A21,A22,FUNCT_1:11;
    let E be Ordinal such that
A23: x c= E and
A24: E in dom fi;
    reconsider E1 = f1.E as Ordinal;
A25: dom fi c= dom f1 by A4,RELAT_1:25;
    then E1 in rng f1 by A24,FUNCT_1:def 3;
    then
A26: E1 in dom f2 by A3,ORDINAL2:19;
    B1 c= E1 by A1,A22,A23,A24,A25,Th9;
    then
A27: D c= E1 by A20;
    then
A28: f2.E1 in C by A18,A26;
    B in f2.E1 by A18,A27,A26;
    hence thesis by A4,A24,A28,FUNCT_1:12;
  end;
end;
