
theorem
for f,Intf be PartFunc of REAL,REAL, b be Real
 st f is_-infty_improper_integrable_on b & dom Intf = left_closed_halfline b
  & (for x be Real st x in dom Intf holds Intf.x = integral(f,x,b))
  & 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 = left_closed_halfline b and
A3:  for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A4:  Intf is convergent_in-infty;

A5: f is_-infty_ext_Riemann_integrable_on b by A1,A2,A3,A4,INTEGR10:def 6;
    then
A6: improper_integral_-infty(f,b) = infty_ext_left_integral(f,b) by A1,Th22;

    consider I be PartFunc of REAL,REAL such that
A7:  dom I = left_closed_halfline b and
A8:  for x be Real st x in dom I holds I.x = integral(f,x,b) and
     I is convergent_in-infty and
A9:  infty_ext_left_integral(f,b) = lim_in-infty I by A5,INTEGR10:def 8;

    now let x be Element of REAL;
     assume A10: x in dom Intf; then
     Intf.x = integral(f,x,b) 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;
