reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f1 is convergent_in+infty & lim_in+infty f1=0 & (for r ex g st r<g & g
in dom(f1(#)f2)) & (ex r st f2|right_open_halfline r is bounded) implies f1(#)
  f2 is convergent_in+infty & lim_in+infty(f1(#)f2)=0
proof
  assume that
A1: f1 is convergent_in+infty & lim_in+infty f1=0 and
A2: for r ex g st r<g & g in dom(f1(#)f2);
  given r such that
A3: f2|right_open_halfline r is bounded;
  consider g be Real such that
A4: for r1 being object st r1 in right_open_halfline(r)/\dom f2 holds |.
  f2.r1.|<=g by A3,RFUNCT_1:73;
A5: now
    let s be Real_Sequence;
    assume that
A6: s is divergent_to+infty and
A7: rng s c=dom(f1(#)f2);
    consider k such that
A8: for n st k<=n holds r<s.n by A6;
A9: rng(s^\k)c=rng s by VALUED_0:21;
A10: rng s c=dom f2 by A7,Lm3;
    then
A11: rng(s^\k)c=dom f2 by A9;
    now
      set t=|.g.|+1;
      0<=|.g.| by COMPLEX1:46;
      hence 0<t;
      let n;
A12: n in NAT by ORDINAL1:def 12;
      r<s.(n+k) by A8,NAT_1:12;
      then r<(s^\k).n by NAT_1:def 3;
      then (s^\k).n in {g1: r<g1};
      then (s^\k).n in rng(s^\k) & (s^\k).n in right_open_halfline(r) by
VALUED_0:28,XXREAL_1:230;
      then (s^\k).n in right_open_halfline(r)/\dom f2 by A11,XBOOLE_0:def 4;
      then |.f2.((s^\k).n).|<=g by A4;
      then
A13:  |.(f2/*(s^\k)).n.|<= g by A10,A9,FUNCT_2:108,XBOOLE_1:1,A12;
      g<=|.g.| by ABSVALUE:4;
      then g<t by Lm1;
      hence |.(f2/*(s^\k)).n.|<t by A13,XXREAL_0:2;
    end;
    then
A14: f2/*(s^\k) is bounded by SEQ_2:3;
    dom(f1(#)f2)=dom f1/\dom f2 by A7,Lm3;
    then rng(s^\k)c=dom f1/\dom f2 by A7,A9;
    then
A15: (f1/*(s^\k))(#)(f2/*(s^\k))=(f1(#)f2)/*(s^\k) by RFUNCT_2:8
      .=((f1(#)f2)/*s)^\k by A7,VALUED_0:27;
    rng s c=dom f1 by A7,Lm3;
    then
A16: rng(s^\k)c=dom f1 by A9;
    s^\k is divergent_to+infty by A6,Th26;
    then
A17: f1/*(s^\k) is convergent & lim(f1/*(s^\k))=0 by A1,A16,Def12;
    then
A18: (f1/*(s^\k))(#)(f2/*(s^\k)) is convergent by A14,SEQ_2:25;
    hence (f1(#)f2)/*s is convergent by A15,SEQ_4:21;
    lim((f1/*(s^\k))(#)(f2/*(s^\k)))=0 by A17,A14,SEQ_2:26;
    hence lim((f1(#)f2)/*s)=0 by A18,A15,SEQ_4:22;
  end;
  hence f1(#) f2 is convergent_in+infty by A2;
  hence thesis by A5,Def12;
end;
