
theorem Th32:
for f be PartFunc of REAL,REAL, a,b be Real
 st f is_left_ext_Riemann_integrable_on a,b
 holds f is_left_improper_integrable_on a,b
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume f is_left_ext_Riemann_integrable_on a,b; then
    (for d be Real st a < d & d <= b holds
      f is_integrable_on [' d,b '] & f|[' d,b '] 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,x,b))
    & Intf is_right_convergent_in a by INTEGR10:def 2;
    hence thesis;
end;
