
theorem Th35:
for f be PartFunc of REAL,REAL, a,b be Real st
  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_divergent_to+infty_in a or
     Intf is_right_divergent_to-infty_in a)
 holds not f is_left_ext_Riemann_integrable_on a,b
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume 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_divergent_to+infty_in a or
       Intf is_right_divergent_to-infty_in a); then
    consider Intf be PartFunc of REAL,REAL such that
A1:     dom Intf = ].a,b.] and
A2:  for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A3:  Intf is_right_divergent_to+infty_in a
    or Intf is_right_divergent_to-infty_in a;

    hereby assume f is_left_ext_Riemann_integrable_on a,b; then
     consider I be PartFunc of REAL,REAL such that
A4:   dom I = ].a,b.] and
A5:   for x be Real st x in dom I holds I.x = integral(f,x,b) and
A6:   I is_right_convergent_in a by INTEGR10:def 2;
     now let x be Element of REAL;
      assume A7: x in dom I; then
      I.x = integral(f,x,b) by A5;
      hence I.x = Intf.x by A1,A4,A7,A2;
     end; then
     I = Intf by A1,A4,PARTFUN1:5;
     hence contradiction by A3,A6,Th8,Th9;
    end;
end;
