reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  (s=s1(#)s2 & for n holds s1.n<0 & s2.n<0) implies (Partial_Sums s).n<=
  (Partial_Sums(s1).n)*(Partial_Sums(s2).n)
proof
  assume that
A1: s=s1(#)s2 and
A2: for n holds s1.n<0 & s2.n<0;
  defpred X[Nat] means (Partial_Sums s).$1<= (Partial_Sums(s1).$1)*
  (Partial_Sums(s2).$1);
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    set j=(Partial_Sums(s)).n;
    set u=Partial_Sums(s1).n;
    set v=Partial_Sums(s2).n;
    set w=s1.(n+1);
    set h=s2.(n+1);
A4: (Partial_Sums(s1).(n+1))*(Partial_Sums(s2).(n+1)) =(Partial_Sums(s1).n
    +s1.(n+1))*(Partial_Sums(s2).(n+1)) by SERIES_1:def 1
      .=(u+w)*(v+h) by SERIES_1:def 1
      .=u*v+u*h+w*v+w*h;
    assume (Partial_Sums(s)).n<= (Partial_Sums(s1).n)*(Partial_Sums(s2).n);
    then
A5: j+w*h<=u*v+w*h by XREAL_1:6;
A6: w<0 & h<0 by A2;
    u<0 & v<0 by A2,Th35;
    then
A7: (u*v+w*h)<=(u*v+w*h)+(u*h+w*v) by A6,XREAL_1:31;
    (Partial_Sums(s)).(n+1) =(Partial_Sums(s)).n+s.(n+1) by SERIES_1:def 1
      .=(Partial_Sums(s)).n+s1.(n+1)*s2.(n+1) by A1,SEQ_1:8;
    hence thesis by A4,A5,A7,XXREAL_0:2;
  end;
A8: (Partial_Sums(s1).0)*(Partial_Sums(s2).0) =s1.0*(Partial_Sums(s2).0) by
SERIES_1:def 1
    .=s1.0*s2.0 by SERIES_1:def 1;
  (Partial_Sums s).0 = s.0 by SERIES_1:def 1
    .=s1.0*s2.0 by A1,SEQ_1:8;
  then
A9: X[0] by A8;
  for n holds X[n] from NAT_1:sch 2(A9,A3);
  hence thesis;
end;
