
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_left_ext_Riemann_integrable_on c,d
  & ext_left_integral(f,c,d) = integral(f,c,d)
proof
   let f be PartFunc of REAL,REAL, a,b be Real;
   assume that
A1: [.a,b.[ c= dom f and
A2: f is_right_ext_Riemann_integrable_on a,b;

   hereby let c,d be Real;
    assume that
A3:  a <= c < d and
A4:  d < b;

A5: a < d by A3,XXREAL_0:2; then
    ['a,d'] = [.a,d.] & ['a,b'] = [.a,b.] & ['c,d'] = [.c,d.]
      by A3,A4,XXREAL_0:2,INTEGRA5:def 3; then
    ['a,d'] c= [.a,b.[ & ['c,d'] c= [.a,b.[ by A4,A3,XXREAL_1:43; then
A6: ['a,d'] c= dom f & ['c,d'] c= dom f by A1;
    f is_integrable_on [' a,d '] & f|[' a,d '] is bounded
       by A4,A5,A2,INTEGR10:def 1; then
A7: f is_integrable_on [' c,d '] & f|[' c,d '] is bounded by A3,A6,INTEGRA6:18;
    hence f is_left_ext_Riemann_integrable_on c,d by A3,A6,Th18;
    thus ext_left_integral(f,c,d) = integral(f,c,d) by A3,A6,A7,Th18;
   end;
end;
