
theorem
for f be PartFunc of REAL,REAL, a,b be Real
 st [.a,b.[ c= dom f & f is_right_ext_Riemann_integrable_on a,b holds
  for c,d be Real st a <= c & c < d & d <= b holds
    f is_right_ext_Riemann_integrable_on c,d
  & (d < b implies ext_right_integral(f,c,d) = integral(f,c,d))
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume A1: [.a,b.[ c= dom f & f is_right_ext_Riemann_integrable_on a,b;
    hereby let c,d be Real;
     assume that
A2:   a <= c and
A3:   c < d and
A4:   d <= b;
     a < d by A2,A3,XXREAL_0:2; then
A5:  f is_right_ext_Riemann_integrable_on a,d by A1,A4,Lm6;

     [.a,d.[ c= [.a,b.[ by A4,XXREAL_1:38; then
     [.a,d.[ c= dom f by A1;
     hence f is_right_ext_Riemann_integrable_on c,d by A2,A3,A5,Th25;
     thus d < b implies ext_right_integral(f,c,d) = integral(f,c,d)
     proof
      assume
A6:    d < b;
       c < b by A3,A4,XXREAL_0:2; then
       f is_right_ext_Riemann_integrable_on c,b by A1,A2,Th25; then
A7:    f is_integrable_on ['c,d'] & f|['c,d'] is bounded
         by A3,A6,INTEGR10:def 1;
       ['c,d'] = [.c,d.] by A3,INTEGRA5:def 3; then
       ['c,d'] c= [.a,b.[ by A2,A6,XXREAL_1:43; then
       ['c,d'] c= dom f by A1;
       hence ext_right_integral(f,c,d) = integral(f,c,d)
         by A3,A7,INTEGR10:12;
     end;
    end;
end;
