
theorem
for f be PartFunc of REAL,REAL st dom f = REAL holds
 f is infty_ext_Riemann_integrable iff
 for a be Real holds f is_+infty_ext_Riemann_integrable_on a
  & f is_-infty_ext_Riemann_integrable_on a
proof
    let f be PartFunc of REAL,REAL;
    assume A1: dom f = REAL;

    hereby assume
A2: f is infty_ext_Riemann_integrable; then
A3:  f is_+infty_ext_Riemann_integrable_on 0
   & f is_-infty_ext_Riemann_integrable_on 0 by INTEGR10:def 9;

     thus for a be Real holds f is_+infty_ext_Riemann_integrable_on a
       & f is_-infty_ext_Riemann_integrable_on a
     proof
      let a be Real;
      per cases;
      suppose A4: 0 <= a;

       right_closed_halfline(0) c= dom f by A1;
       hence f is_+infty_ext_Riemann_integrable_on a
         by A2,A4,Th16,INTEGR10:def 9;

A5:    left_closed_halfline(a) c= dom f by A1;
       f is_integrable_on ['0,a'] & f|['0,a'] is bounded
         by A3,A4,INTEGR10:def 5;
       hence f is_-infty_ext_Riemann_integrable_on a by A3,A4,A5,Th17;
      end;
      suppose A6: a < 0;

A7:    right_closed_halfline(a) c= dom f by A1;
       f is_integrable_on ['a,0'] & f|['a,0'] is bounded
         by A3,A6,INTEGR10:def 6;
       hence f is_+infty_ext_Riemann_integrable_on a by A3,A6,A7,Th18;
       left_closed_halfline(0) c= dom f by A1;
       hence f is_-infty_ext_Riemann_integrable_on a
         by A2,A6,Th15,INTEGR10:def 9;
      end;
     end;
    end;

    assume for a be Real holds f is_+infty_ext_Riemann_integrable_on a
       & f is_-infty_ext_Riemann_integrable_on a; then
    f is_+infty_ext_Riemann_integrable_on 0 &
    f is_-infty_ext_Riemann_integrable_on 0;
    hence f is infty_ext_Riemann_integrable by INTEGR10:def 9;
end;
