
theorem Th44:
for f be PartFunc of REAL,REAL, a be Real st right_closed_halfline a c= dom f &
 f is_+infty_improper_integrable_on a holds
   -f is_+infty_improper_integrable_on a &
   improper_integral_+infty(-f,a) = - improper_integral_+infty(f,a)
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume A1: right_closed_halfline a c= dom f
      & f is_+infty_improper_integrable_on a;
    hence -f is_+infty_improper_integrable_on a by Th42;
    improper_integral_+infty(-f,a) = (-1) * improper_integral_+infty(f,a)
      by A1,Th42;
    hence improper_integral_+infty(-f,a) = - improper_integral_+infty(f,a)
      by XXREAL_3:91;
end;
