
theorem Th42:
for f be PartFunc of REAL,REAL, a,b,c be Real st a <= b < c & [.a,c.[ c= dom f
 & f is_right_improper_integrable_on a,c holds
  f is_right_improper_integrable_on b,c &
  ( right_improper_integral(f,a,c) = ext_right_integral(f,a,c)
     implies right_improper_integral(f,b,c) = ext_right_integral(f,b,c) ) &
  ( right_improper_integral(f,a,c) = +infty
     implies right_improper_integral(f,b,c) = +infty ) &
  ( right_improper_integral(f,a,c) = -infty
     implies right_improper_integral(f,b,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 is_right_improper_integrable_on a,c;

    per cases;
    suppose f is_right_ext_Riemann_integrable_on a,c; then
     right_improper_integral(f,a,c) = ext_right_integral(f,a,c) by A3,Th39;
     hence thesis by A1,A2,A3,Lm16;
    end;
    suppose not f is_right_ext_Riemann_integrable_on a,c; then
     per cases by A3,Th39;
     suppose right_improper_integral(f,a,c) = +infty;
      hence thesis by A1,A2,A3,Lm17;
     end;
     suppose right_improper_integral(f,a,c) = -infty;
      hence thesis by A1,A2,A3,Lm18;
     end;
    end;
end;
