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 Th13:
  f1 is increasing & f2 is increasing implies ex phi st phi = f1*
  f2 & phi is increasing
proof
  assume that
A1: f1 is increasing and
A2: f2 is increasing;
  reconsider f = f1*f2 as Ordinal-Sequence by A2,Th12;
  take f;
  thus f = f1*f2;
  let A,B;
  assume that
A3: A in B and
A4: B in dom f;
  reconsider A1 = f2.A, B1 = f2.B as Ordinal;
A5: B1 in dom f1 by A4,FUNCT_1:11;
  dom f c= dom f2 by RELAT_1:25;
  then
A6: A1 in B1 by A2,A3,A4;
A7: f.B = f1.B1 by A4,FUNCT_1:12;
  f.A = f1.A1 by A3,A4,FUNCT_1:12,ORDINAL1:10;
  hence thesis by A1,A6,A5,A7;
end;
