
theorem Th35:
for f be PartFunc of REAL,REAL st dom f = REAL
 & f is_improper_integrable_on_REAL holds
for b1,b2 be Real holds
 improper_integral_-infty(f,b1)+improper_integral_+infty(f,b1)
  = improper_integral_-infty(f,b2)+improper_integral_+infty(f,b2)
proof
    let f be PartFunc of REAL,REAL;
    assume that
A1:  dom f = REAL and
A2:  f is_improper_integrable_on_REAL;
    let b1,b2 be Real;

    consider b be Real such that
A3:  f is_-infty_improper_integrable_on b and
A4:  f is_+infty_improper_integrable_on b and
A5:  not (improper_integral_-infty(f,b) = -infty
       & improper_integral_+infty(f,b) = +infty) and
A6:  not (improper_integral_-infty(f,b) = +infty
       & improper_integral_+infty(f,b) = -infty) by A2;

    per cases;
    suppose b1 < b & b2 < b; then
     improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b1)+improper_integral_+infty(f,b1)
   & improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b2)+improper_integral_+infty(f,b2)
       by A1,A3,A4,A5,A6,Th33;
     hence thesis;
    end;
    suppose b1 < b & b <= b2; then
     improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b1)+improper_integral_+infty(f,b1)
   & improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b2)+improper_integral_+infty(f,b2)
       by A1,A3,A4,A5,A6,Th33,Th34;
     hence thesis;
    end;
    suppose b <= b1 & b2 < b; then
     improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b1)+improper_integral_+infty(f,b1)
   & improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b2)+improper_integral_+infty(f,b2)
       by A1,A3,A4,A5,A6,Th33,Th34;
     hence thesis;
    end;
    suppose b <= b1 & b <= b2; then
     improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b1)+improper_integral_+infty(f,b1)
   & improper_integral_-infty(f,b)+improper_integral_+infty(f,b)
      = improper_integral_-infty(f,b2)+improper_integral_+infty(f,b2)
       by A1,A3,A4,A5,A6,Th34;
     hence thesis;
    end;
end;
