
theorem Th25:
for f be PartFunc of REAL,REAL, b,c be Real
 st b <= c & left_closed_halfline c c= dom f
  & f is_-infty_improper_integrable_on c
 holds f is_-infty_improper_integrable_on b
  & ( improper_integral_-infty(f,c) = infty_ext_left_integral(f,c)
     implies improper_integral_-infty(f,b) = infty_ext_left_integral(f,b) )
  & ( improper_integral_-infty(f,c) = +infty
     implies improper_integral_-infty(f,b) = +infty )
  & ( improper_integral_-infty(f,c) = -infty
     implies improper_integral_-infty(f,b) = -infty )
proof
    let f be PartFunc of REAL,REAL, b,c be Real;
    assume that
A1:  b <= c and
A2:  left_closed_halfline c c= dom f and
A3:  f is_-infty_improper_integrable_on c;

    per cases;
    suppose f is_-infty_ext_Riemann_integrable_on c; then
     improper_integral_-infty(f,c) = infty_ext_left_integral(f,c)
       by A3,Th22;
     hence thesis by A1,A2,A3,Lm3;
    end;
    suppose not f is_-infty_ext_Riemann_integrable_on c; then
     per cases by A3,Th22;
     suppose improper_integral_-infty(f,c) = +infty;
      hence thesis by A1,A2,A3,Lm4;
     end;
     suppose improper_integral_-infty(f,c) = -infty;
      hence thesis by A1,A2,A3,Lm5;
     end;
    end;
end;
