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
  fi is increasing & fi is continuous & psi is continuous & phi = psi*fi
  implies phi is continuous
proof
  assume that
A1: fi is increasing and
A2: for A,B st A in dom fi & A <> 0 & A is limit_ordinal & B = fi.A
  holds B is_limes_of fi|A and
A3: for A,B st A in dom psi & A <> 0 & A is limit_ordinal & B = psi.A
  holds B is_limes_of psi|A and
A4: phi = psi*fi;
  let A,B such that
A5: A in dom phi and
A6: A <> 0 and
A7: A is limit_ordinal and
A8: B = phi.A;
  reconsider A1 = fi.A as Ordinal;
A9: fi|A is increasing by A1,Th15;
A10: dom phi c= dom fi by A4,RELAT_1:25;
  then A c= dom fi by A5,ORDINAL1:def 2;
  then
A11: dom (fi|A) = A by RELAT_1:62;
  A1 is_limes_of fi|A by A2,A5,A6,A7,A10;
  then lim (fi|A) = A1 by ORDINAL2:def 10;
  then
A12: sup (fi|A) = A1 by A6,A7,A11,A9,Th8;
A13: B = psi.A1 by A4,A5,A8,FUNCT_1:12;
A14: {} in A by A6,ORDINAL3:8;
A15: A1 in dom psi by A4,A5,FUNCT_1:11;
  then A1 c= dom psi by ORDINAL1:def 2;
  then
A16: dom (psi|A1) = A1 by RELAT_1:62;
A17: rng (fi|A) c= sup rng (fi|A)
  proof
    let x be object;
    assume
A18: x in rng (fi|A);
    then ex y being object st y in dom (fi|A) & x = (fi|A).y by FUNCT_1:def 3;
    hence thesis by A18,ORDINAL2:19;
  end;
  phi|A = psi*(fi| A ) by A4,RELAT_1:83;
  then
A19: phi|A = (psi|A1)*(fi|A) by A17,A12,Lm5;
  A c= A1 by A1,A5,A10,Th10;
  then B is_limes_of psi|A1 by A3,A7,A13,A15,A11,A14,A9,A12,Th16;
  hence thesis by A9,A16,A12,A19,Th14;
end;
