reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f1 is_divergent_to+infty_in x0 & (for r1,r2 st r1<x0 & x0<r2 ex g1,g2
st r1<g1 & g1<x0 & g1 in dom(f1(#)f2) & g2<r2 & x0<g2 & g2 in dom(f1(#)f2)) & (
  ex r,r1 st 0<r & 0<r1 & for g st g in dom f2 /\ (].x0-r,x0.[ \/ ].x0,x0+r.[)
  holds r1<=f2.g) implies f1(#)f2 is_divergent_to+infty_in x0
proof
  assume that
A1: f1 is_divergent_to+infty_in x0 and
A2: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f1
  (#)f2) & g2<r2 & x0<g2 & g2 in dom(f1(#)f2);
  given r,t such that
A3: 0<r and
A4: 0<t and
A5: for g st g in dom f2/\(].x0-r,x0.[\/ ].x0,x0+r.[) holds t<=f2.g;
  now
    let s be Real_Sequence;
    assume that
A6: s is convergent and
A7: lim s=x0 and
A8: rng s c=dom(f1(#)f2)\{x0};
    consider k such that
A9: for n st k<=n holds x0-r<s.n & s.n<x0+r by A3,A6,A7,Th7;
A10: rng s c=dom(f1(#)f2) by A8,Lm2;
A11: dom(f1(#)f2)=dom f1/\dom f2 by A8,Lm2;
    rng(s^\k)c=rng s by VALUED_0:21;
    then
A12: rng(s^\k)c=dom(f1(#)f2)\{x0} by A8;
    then
A13: rng(s^\k)c=dom f1\{x0} by Lm2;
A14: rng(s^\k)c=dom f2 by A12,Lm2;
A15: now
      thus 0<t by A4;
      let n be Nat;
A16:    n in NAT by ORDINAL1:def 12;
A17:  k<=n+k by NAT_1:12;
      then s.(n+k)<x0+r by A9;
      then
A18:  (s^\k).n<x0+r by NAT_1:def 3;
      x0-r<s.(n+k) by A9,A17;
      then x0-r<(s^\k).n by NAT_1:def 3;
      then (s^\k).n in {g2: x0-r<g2 & g2<x0+r} by A18;
      then
A19:  (s^\k).n in ].x0-r,x0+r.[ by RCOMP_1:def 2;
A20:  (s^\k).n in rng(s ^\k) by VALUED_0:28;
      then not (s^\k).n in {x0} by A12,XBOOLE_0:def 5;
      then (s^\k).n in ].x0-r,x0+r.[\{x0} by A19,XBOOLE_0:def 5;
      then (s^\k).n in ].x0-r,x0.[\/].x0,x0+r.[ by A3,Th4;
      then (s^\k).n in dom f2/\(].x0-r,x0.[\/].x0,x0+r.[) by A14,A20,
XBOOLE_0:def 4;
      then t<=f2.((s^\k).n) by A5;
      hence t<=(f2/*(s^\k)).n by A14,FUNCT_2:108,A16;
    end;
    lim(s^\k)=x0 by A6,A7,SEQ_4:20;
    then f1/*(s^\k) is divergent_to+infty by A1,A6,A13;
    then
A21: (f1/*(s^\k))(#)(f2/*(s^\k)) is divergent_to+infty by A15,LIMFUNC1:22;
    rng(s^\k)c=dom(f1(#)f2) by A12,Lm2;
    then (f1/*(s^\k))(#)(f2/*(s^\k))=(f1(#)f2)/*(s^\k) by A11,RFUNCT_2:8
      .=((f1(#)f2)/*s)^\k by A10,VALUED_0:27;
    hence (f1(#)f2)/*s is divergent_to+infty by A21,LIMFUNC1:7;
  end;
  hence thesis by A2;
end;
