
theorem
for f be PartFunc of REAL,REAL, d be Real st
 left_closed_halfline(d) c= dom f & f is_-infty_ext_Riemann_integrable_on d
 holds for b,c be Real st b < c <= d 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, d be Real;
    assume that
A1:  left_closed_halfline(d) c= dom f and
A2:  f is_-infty_ext_Riemann_integrable_on d;

    hereby let b,c be Real;
     assume A3: b < c <= d; then
     b < d by XXREAL_0:2; then
A4:  f is_integrable_on ['b,d'] & f|['b,d'] is bounded by A2,INTEGR10:def 6;
     ['b,d'] = [.b,d.] by A3,XXREAL_0:2,INTEGRA5:def 3; then
     ['b,d'] c= ].-infty,d.] by XXREAL_1:265; then
     ['b,d'] 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;
