
theorem Th38:
for f be PartFunc of REAL,REAL, b be Real st
 f is_-infty_improper_integrable_on b &
 improper_integral_-infty(f,b) = -infty holds
 for Intf be PartFunc of REAL,REAL st dom Intf = left_closed_halfline b &
  (for x be Real st x in dom Intf holds Intf.x = integral(f,x,b)) holds
  Intf is divergent_in-infty_to-infty
proof
    let f be PartFunc of REAL,REAL, b be Real;
    assume that
A1:  f is_-infty_improper_integrable_on b and
A2:  improper_integral_-infty(f,b) = -infty;
    let Intf be PartFunc of REAL,REAL;
    assume that
A3:  dom Intf = left_closed_halfline b and
A4:  for x be Real st x in dom Intf holds Intf.x = integral(f,x,b);

    consider I be PartFunc of REAL,REAL such that
A5:  dom I = left_closed_halfline b &
     (for x be Real st x in dom I holds I.x = integral(f,x,b))
   & (I is convergent_in-infty
       & improper_integral_-infty(f,b) = lim_in-infty I
   or I is divergent_in-infty_to+infty
       & improper_integral_-infty(f,b) = +infty
   or I is divergent_in-infty_to-infty
       & improper_integral_-infty(f,b) = -infty) by A1,Def3;

    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,x,b) & Intf.x = integral(f,x,b) by A3,A4,A5;
     hence thesis;
    end;
    hence Intf is divergent_in-infty_to-infty by A2,A3,A5,PARTFUN1:5;
end;
