
theorem
for f be PartFunc of REAL,REAL, a be Real st
 right_closed_halfline(a) c= dom f & f is_+infty_ext_Riemann_integrable_on a
  holds for b,c be Real st a <= b < c holds
    f is_right_ext_Riemann_integrable_on b,c
  & f is_left_ext_Riemann_integrable_on b,c
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1: right_closed_halfline(a) c= dom f and
A2: f is_+infty_ext_Riemann_integrable_on a;

    hereby let b,c be Real;
     assume A3: a <= b < c; then
     a < c by XXREAL_0:2; then
A4:  f is_integrable_on ['a,c'] & f|['a,c'] is bounded by A2,INTEGR10:def 5;
     ['a,c'] = [.a,c.] by A3,XXREAL_0:2,INTEGRA5:def 3; then
     ['a,c'] c= [.a,+infty.[ by XXREAL_1:251; then
     ['a,c'] c= dom f by A1; then
     f is_integrable_on ['b,c'] & f|['b,c'] is bounded & ['b,c'] c= dom f
       by A3,A4,INTEGRA6:18;
     hence f is_right_ext_Riemann_integrable_on b,c &
       f is_left_ext_Riemann_integrable_on b,c by A3,INTEGR24:19,18;
    end;
end;
