
theorem Th34:
for f be PartFunc of REAL,REAL, a,b be Real st
 f is_left_improper_integrable_on a,b holds
   (f is_left_ext_Riemann_integrable_on a,b &
    left_improper_integral(f,a,b) = ext_left_integral(f,a,b))
or (not f is_left_ext_Riemann_integrable_on a,b &
    left_improper_integral(f,a,b) = +infty)
or (not f is_left_ext_Riemann_integrable_on a,b &
    left_improper_integral(f,a,b) = -infty)
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume
A1:  f is_left_improper_integrable_on a,b; then
    consider Intf be PartFunc of REAL,REAL such that
A2:  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 or Intf is_right_divergent_to+infty_in a
     or Intf is_right_divergent_to-infty_in a);
    per cases by A2;
    suppose A3: Intf is_right_convergent_in a; then
A4:  f is_left_ext_Riemann_integrable_on a,b by A1,A2,INTEGR10:def 2;
     left_improper_integral(f,a,b) = lim_right(Intf,a) by A1,A2,A3,Def3
      .= ext_left_integral(f,a,b) by A2,A3,A4,INTEGR10:def 4;
     hence thesis by A1,A2,A3,INTEGR10:def 2;
    end;
    suppose A5: Intf is_right_divergent_to+infty_in a;
     for I be PartFunc of REAL,REAL st
      dom I = ].a,b.]
    & (for x be Real st x in dom I holds I.x = integral(f,x,b))
     holds not I is_right_convergent_in a
     proof
      let I be PartFunc of REAL,REAL;
      assume that
A6:    dom I = ].a,b.] and
A7:    for x be Real st x in dom I holds I.x = integral(f,x,b);
      now let x be Element of REAL;
       assume A8: x in dom Intf; then
       Intf.x = integral(f,x,b) by A2;
       hence Intf.x = I.x by A2,A6,A7,A8;
      end; then
      Intf = I by A2,A6,PARTFUN1:5;
      hence not I is_right_convergent_in a by A5,Th8;
     end;
     hence thesis by A1,Def3,INTEGR10:def 2;
    end;
    suppose A9: Intf is_right_divergent_to-infty_in a;
     for I be PartFunc of REAL,REAL st
      dom I = ].a,b.]
    & (for x be Real st x in dom I holds I.x = integral(f,x,b))
     holds not I is_right_convergent_in a
     proof
      let I be PartFunc of REAL,REAL;
      assume that
A10:   dom I = ].a,b.] and
A11:   for x be Real st x in dom I holds I.x = integral(f,x,b);
      now let x be Element of REAL;
       assume A12: x in dom Intf; then
       Intf.x = integral(f,x,b) by A2;
       hence Intf.x = I.x by A2,A10,A11,A12;
      end; then
      Intf = I by A2,A10,PARTFUN1:5;
      hence not I is_right_convergent_in a by A9,Th9;
     end;
     hence thesis by A1,Def3,INTEGR10:def 2;
    end;
end;
