
theorem Th50:
for f be PartFunc of REAL,REAL, r be Real st dom f = REAL &
 f is_improper_integrable_on_REAL holds
  -f is_improper_integrable_on_REAL &
  improper_integral_on_REAL -f = - improper_integral_on_REAL f
proof
    let f be PartFunc of REAL,REAL, r be Real;
    assume dom f = REAL & f is_improper_integrable_on_REAL; then
    (-1)(#)f is_improper_integrable_on_REAL &
    improper_integral_on_REAL ((-1)(#)f) = (-1) * improper_integral_on_REAL f
      by Th49;
    hence thesis by XXREAL_3:91;
end;
