
theorem Th43:
for f be PartFunc of REAL,REAL, a,b,c be Real
 st a <= b < c & [.a,c.[ c= dom f & f|['a,b'] is bounded
  & f is_right_improper_integrable_on b,c & f is_integrable_on ['a,b']
holds f is_right_improper_integrable_on a,c &
  ( right_improper_integral(f,b,c) = ext_right_integral(f,b,c) implies
     right_improper_integral(f,a,c)
     = right_improper_integral(f,b,c) + integral(f,a,b) ) &
  ( right_improper_integral(f,b,c) = +infty implies
     right_improper_integral(f,a,c) = +infty ) &
  ( right_improper_integral(f,b,c) = -infty implies
     right_improper_integral(f,a,c) = -infty )
proof
    let f be PartFunc of REAL,REAL, a,b,c be Real;
    assume that
A1:  a <= b < c and
A2:  [.a,c.[ c= dom f and
A3:  f|['a,b'] is bounded and
A4:  f is_right_improper_integrable_on b,c and
A5:  f is_integrable_on ['a,b'];
    per cases;
    suppose f is_right_ext_Riemann_integrable_on b,c; then
     right_improper_integral(f,b,c) = ext_right_integral(f,b,c) by A4,Th39;
     hence thesis by A1,A2,A3,A4,A5,Lm20;
    end;
    suppose not f is_right_ext_Riemann_integrable_on b,c; then
     per cases by A4,Th39;
     suppose right_improper_integral(f,b,c) = +infty;
      hence thesis by A1,A2,A3,A4,A5,Lm21;
     end;
     suppose right_improper_integral(f,b,c) = -infty;
      hence thesis by A1,A2,A3,A4,A5,Lm22;
     end;
    end;
end;
