
theorem
for f be PartFunc of REAL,REAL, a,b be Real
 st ].a,b.] c= dom f & f is_left_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
  & (a < c implies ext_left_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_left_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;
     c < b by A3,A4,XXREAL_0:2; then

A5:  f is_left_ext_Riemann_integrable_on c,b by A1,A2,Lm4;
     ].c,b.] c= ].a,b.] by A2,XXREAL_1:42; then
     ].c,b.] c= dom f by A1;
     hence f is_left_ext_Riemann_integrable_on c,d by A3,A4,A5,Th22;
     thus a < c implies ext_left_integral(f,c,d) = integral(f,c,d)
     proof
      assume
A6:    a < c;
       a < d by A2,A3,XXREAL_0:2; then
       f is_left_ext_Riemann_integrable_on a,d by A1,A4,Th22; then
A7:    f is_integrable_on ['c,d'] & f|['c,d'] is bounded
         by A3,A6,INTEGR10:def 2;
       ['c,d'] = [.c,d.] by A3,INTEGRA5:def 3; then
       ['c,d'] c= ].a,b.] by A4,A6,XXREAL_1:39; then
       ['c,d'] c= dom f by A1;
       hence ext_left_integral(f,c,d) = integral(f,c,d)
         by A3,A7,INTEGR10:13;
     end;
    end;
end;
