
theorem Th33:
for f be PartFunc of REAL,REAL, a,b be Real
 st f is_right_ext_Riemann_integrable_on a,b
 holds f is_right_improper_integrable_on a,b
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume f is_right_ext_Riemann_integrable_on a,b; then
    (for d be Real st a <= d & d < b holds
      f is_integrable_on [' a,d '] & f|[' a,d '] is bounded) &
    ex Intf be PartFunc of REAL,REAL st dom Intf = [.a,b.[ &
      (for x be Real st x in dom Intf holds Intf.x = integral(f,a,x))
    & Intf is_left_convergent_in b by INTEGR10:def 1;
    hence thesis;
end;
