
theorem
for f,Intf be PartFunc of REAL,REAL, b be Real
 st f is_+infty_improper_integrable_on b & dom Intf = right_closed_halfline b
  & (for x be Real st x in dom Intf holds Intf.x = integral(f,b,x))
  & Intf is convergent_in+infty
holds improper_integral_+infty(f,b) = lim_in+infty Intf
proof
    let f,Intf be PartFunc of REAL,REAL, b be Real;
    assume that
A1:  f is_+infty_improper_integrable_on b and
A2:  dom Intf = right_closed_halfline b and
A3:  for x be Real st x in dom Intf holds Intf.x = integral(f,b,x) and
A4:  Intf is convergent_in+infty;

A5: f is_+infty_ext_Riemann_integrable_on b by A1,A2,A3,A4,INTEGR10:def 5;
    then
A6: improper_integral_+infty(f,b) = infty_ext_right_integral(f,b)
      by A1,Th27;

    consider I be PartFunc of REAL,REAL such that
A7:  dom I = right_closed_halfline b and
A8:  for x be Real st x in dom I holds I.x = integral(f,b,x) and
     I is convergent_in+infty and
A9:  infty_ext_right_integral(f,b) = lim_in+infty I by A5,INTEGR10:def 7;

    now let x be Element of REAL;
     assume A10: x in dom Intf; then
     Intf.x = integral(f,b,x) by A3;
     hence Intf.x = I.x by A2,A7,A8,A10;
    end;
    hence improper_integral_+infty(f,b) = lim_in+infty Intf
      by A6,A9,A2,A7,PARTFUN1:5;
end;
