
theorem Th26:
for f be PartFunc of REAL,REAL, b,c be Real
 st b <= c & left_closed_halfline c c= dom f & f|['b,c'] is bounded
  & f is_-infty_improper_integrable_on b & f is_integrable_on ['b,c']
holds f is_-infty_improper_integrable_on c
 & ( improper_integral_-infty(f,b) = infty_ext_left_integral(f,b) implies
    improper_integral_-infty(f,c)
     = improper_integral_-infty(f,b) + integral(f,b,c) )
 & ( improper_integral_-infty(f,b) = +infty implies
     improper_integral_-infty(f,c) = +infty )
 & ( improper_integral_-infty(f,b) = -infty implies
     improper_integral_-infty(f,c) = -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|['b,c'] is bounded and
A4:  f is_-infty_improper_integrable_on b and
A5:  f is_integrable_on ['b,c'];
    per cases;
    suppose f is_-infty_ext_Riemann_integrable_on b; then
     improper_integral_-infty(f,b) = infty_ext_left_integral(f,b)
       by A4,Th22;
     hence thesis by A1,A2,A3,A4,A5,Lm7;
    end;
    suppose A6: not f is_-infty_ext_Riemann_integrable_on b;
     per cases by A4,A6,Th22;
     suppose improper_integral_-infty(f,b) = +infty;
      hence thesis by A1,A2,A3,A4,A5,Lm8;
     end;
     suppose improper_integral_-infty(f,b) = -infty;
      hence thesis by A1,A2,A3,A4,A5,Lm9;
     end;
    end;
end;
