
theorem
for f,Intf be PartFunc of REAL,REAL, a,b be Real st
 f is_left_improper_integrable_on a,b &
 dom Intf = ].a,b.] &
 (for x be Real st x in dom Intf holds Intf.x = integral(f,x,b)) &
 Intf is_right_convergent_in a
holds left_improper_integral(f,a,b) = lim_right(Intf,a)
proof
    let f,Intf be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  f is_left_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,x,b) and
A4:  Intf is_right_convergent_in a;

A5: f is_left_ext_Riemann_integrable_on a,b by A1,A2,A3,A4,INTEGR10:def 2;
    then
A6: left_improper_integral(f,a,b) = ext_left_integral(f,a,b) by A1,Th34;

    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,x,b) and
     I is_right_convergent_in a and
A9:  ext_left_integral(f,a,b) = lim_right(I,a) by A5,INTEGR10:def 4;

    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 left_improper_integral(f,a,b) = lim_right(Intf,a)
      by A6,A9,A2,A7,PARTFUN1:5;
end;
