
theorem
for f,Intf be PartFunc of REAL,REAL, a,b be Real st
 f is_right_improper_integrable_on a,b &
 dom Intf = [.a,b.[ &
 (for x be Real st x in dom Intf holds Intf.x = integral(f,a,x)) &
 Intf is_left_convergent_in b
holds right_improper_integral(f,a,b) = lim_left(Intf,b)
proof
    let f,Intf be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  f is_right_improper_integrable_on a,b and
A2:  dom Intf = [.a,b.[ and
A3:  for x be Real st x in dom Intf holds Intf.x = integral(f,a,x) and
A4:  Intf is_left_convergent_in b;

A5: f is_right_ext_Riemann_integrable_on a,b by A1,A2,A3,A4,INTEGR10:def 1;
    then
A6: right_improper_integral(f,a,b) = ext_right_integral(f,a,b) by A1,Th39;

    consider I be PartFunc of REAL,REAL such that
A7:  dom I = [.a,b.[ and
A8:  for x be Real st x in dom I holds I.x = integral(f,a,x) and
     I is_left_convergent_in b and
A9:  ext_right_integral(f,a,b) = lim_left(I,b) by A5,INTEGR10:def 3;

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