
theorem Th43:
for f be PartFunc of REAL,REAL, b be Real st left_closed_halfline b c= dom f &
 f is_-infty_improper_integrable_on b holds
   -f is_-infty_improper_integrable_on b &
   improper_integral_-infty(-f,b) = - improper_integral_-infty(f,b)
proof
    let f be PartFunc of REAL,REAL, b be Real;
    assume A1: left_closed_halfline b c= dom f
    & f is_-infty_improper_integrable_on b;
    hence -f is_-infty_improper_integrable_on b by Th41;
    improper_integral_-infty(-f,b) = (-1) * improper_integral_-infty(f,b)
      by A1,Th41;
    hence improper_integral_-infty(-f,b) = - improper_integral_-infty(f,b)
      by XXREAL_3:91;
end;
