
theorem Th36:
for f be PartFunc of REAL,REAL, b be Real
 st dom f = REAL & f is_improper_integrable_on_REAL
  holds f is_-infty_improper_integrable_on b
   & f is_+infty_improper_integrable_on b
   & improper_integral_on_REAL f
   = improper_integral_-infty(f,b) + improper_integral_+infty(f,b)
proof
    let f be PartFunc of REAL,REAL, b be Real;
    assume that
A1:  dom f = REAL and
A2:  f is_improper_integrable_on_REAL;

    consider b1 be Real such that
A3:  f is_-infty_improper_integrable_on b1 and
A4:  f is_+infty_improper_integrable_on b1 and
A5:  improper_integral_on_REAL f
      = improper_integral_-infty(f,b1) + improper_integral_+infty(f,b1)
        by A1,A2,Def6;

    per cases;
    suppose A6: b < b1;
     hence f is_-infty_improper_integrable_on b by A3,A1,Th25;
A7: right_closed_halfline b c= dom f by A1;
     f|['b,b1'] is bounded & f is_integrable_on ['b,b1'] by A3,A6;
     hence f is_+infty_improper_integrable_on b by A6,A4,A7,Th31;
     thus thesis by A1,A2,A5,Th35;
    end;
    suppose A8: b >= b1;
A9: left_closed_halfline b c= dom f by A1;
     f|['b1,b'] is bounded & f is_integrable_on ['b1,b'] by A4,A8;
     hence f is_-infty_improper_integrable_on b by A3,A8,A9,Th26;
     thus f is_+infty_improper_integrable_on b by A8,A1,A4,Th30;
     thus thesis by A1,A2,A5,Th35;
    end;
end;
