
theorem Th23:
for f be PartFunc of REAL,REAL, b be Real
 st ex Intf be PartFunc of REAL,REAL st dom Intf = left_closed_halfline b
  & (for x be Real st x in dom Intf holds Intf.x = integral(f,x,b))
  & (Intf is divergent_in-infty_to+infty
   or Intf is divergent_in-infty_to-infty)
 holds not f is_-infty_ext_Riemann_integrable_on b
proof
    let f be PartFunc of REAL,REAL, b be Real;
    assume ex Intf be PartFunc of REAL,REAL st
     dom Intf = left_closed_halfline b &
     (for x be Real st x in dom Intf holds Intf.x = integral(f,x,b)) &
     (Intf is divergent_in-infty_to+infty
   or Intf is divergent_in-infty_to-infty); then
    consider Intf be PartFunc of REAL,REAL such that
A1:  dom Intf = left_closed_halfline b and
A2:  for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A3:  Intf is divergent_in-infty_to+infty
  or Intf is divergent_in-infty_to-infty;

    hereby assume f is_-infty_ext_Riemann_integrable_on b; then
     consider I be PartFunc of REAL,REAL such that
A4:   dom I = left_closed_halfline b and
A5:   for x be Real st x in dom I holds I.x = integral(f,x,b) and
A6:   I is convergent_in-infty by INTEGR10:def 6;
     now let x be Element of REAL;
      assume A7: x in dom Intf; then
      Intf.x = integral(f,x,b) by A2;
      hence Intf.x = I.x by A1,A4,A5,A7;
     end; then
     Intf = I by A1,A4,PARTFUN1:5;
     hence contradiction by A3,A6,Th1,Th2;
    end;
end;
