
theorem Th52:
for f be PartFunc of REAL,REAL, a,b be Real st
 f is_right_improper_integrable_on a,b &
 right_improper_integral(f,a,b) = -infty
  holds for Intf be PartFunc of REAL,REAL st dom Intf = [.a,b.[ &
  (for x be Real st x in dom Intf holds Intf.x = integral(f,a,x)) holds
  Intf is_left_divergent_to-infty_in b
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  f is_right_improper_integrable_on a,b and
A2:  right_improper_integral(f,a,b) = -infty;
    let Intf be PartFunc of REAL,REAL;
    assume that
A3:  dom Intf = [.a,b.[ 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 = [.a,b.[ &
     (for x be Real st x in dom I holds I.x = integral(f,a,x))
   & (I is_left_convergent_in b
       & right_improper_integral(f,a,b) = lim_left(I,b)
   or I is_left_divergent_to+infty_in b
       & right_improper_integral(f,a,b) = +infty
   or I is_left_divergent_to-infty_in b
       & right_improper_integral(f,a,b) = -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_left_divergent_to-infty_in b by A2,A3,A5,PARTFUN1:5;
end;
