
theorem Th78:
for f be PartFunc of REAL,REAL, a be Real, A be non empty Subset of REAL
 st right_closed_halfline a c= dom f & A = right_closed_halfline a
  & f is_+infty_improper_integrable_on a
  & abs f is_+infty_ext_Riemann_integrable_on a & f is nonnegative
 holds f|A is_integrable_on L-Meas
     & improper_integral_+infty(f,a) = Integral(L-Meas,f|A)
proof
    let f be PartFunc of REAL,REAL, a be Real, A be non empty Subset of REAL;
    assume that
A1:  right_closed_halfline a c= dom f and
A2:  A = right_closed_halfline a and
A3:  f is_+infty_improper_integrable_on a and
A4:  abs f is_+infty_ext_Riemann_integrable_on a and
A5:  f is nonnegative;

    f is_+infty_ext_Riemann_integrable_on a by A1,A3,A4,Th61;
    hence f|A is_integrable_on L-Meas &
    improper_integral_+infty(f,a) = Integral(L-Meas,f|A) by A1,A2,A3,A5,Th49;
end;
