
theorem Th55:
for f be PartFunc of REAL,REAL, a,b be Real st a < b & ].a,b.] c= dom f &
 f is_left_improper_integrable_on a,b holds
   -f is_left_improper_integrable_on a,b &
   left_improper_integral(-f,a,b) = - left_improper_integral(f,a,b)
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume A1: a < b & ].a,b.] c= dom f & f is_left_improper_integrable_on a,b;
    hence -f is_left_improper_integrable_on a,b by Th53;
    left_improper_integral(-f,a,b) = (-1) * left_improper_integral(f,a,b)
      by A1,Th53;
    hence left_improper_integral(-f,a,b) = - left_improper_integral(f,a,b)
      by XXREAL_3:91;
end;
