
theorem Th40:
for f be PartFunc of REAL,REAL, a be Real st
 f is_+infty_improper_integrable_on a &
 improper_integral_+infty(f,a) = -infty holds
 for Intf be PartFunc of REAL,REAL st dom Intf = right_closed_halfline a &
  (for x be Real st x in dom Intf holds Intf.x = integral(f,a,x)) holds
  Intf is divergent_in+infty_to-infty
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  f is_+infty_improper_integrable_on a and
A2:  improper_integral_+infty(f,a) = -infty;
    let Intf be PartFunc of REAL,REAL;
    assume that
A3:  dom Intf = right_closed_halfline a and
A4:  for x be Real st x in dom Intf holds Intf.x = integral(f,a,x);

    consider I be PartFunc of REAL,REAL such that
A5:  dom I = right_closed_halfline a &
     (for x be Real st x in dom I holds I.x = integral(f,a,x))
   & (I is convergent_in+infty
       & improper_integral_+infty(f,a) = lim_in+infty I
   or I is divergent_in+infty_to+infty
       & improper_integral_+infty(f,a) = +infty
   or I is divergent_in+infty_to-infty
       & improper_integral_+infty(f,a) = -infty) by A1,Def4;

    for x be Element of REAL st x in dom I holds I.x = Intf.x
    proof
     let x be Element of REAL;
     assume x in dom I; then
     I.x = integral(f,a,x) & Intf.x = integral(f,a,x) by A3,A4,A5;
     hence thesis;
    end;
    hence Intf is divergent_in+infty_to-infty by A2,A3,A5,PARTFUN1:5;
end;
