
theorem
for f,g be PartFunc of REAL,REAL, a be Real
 st right_closed_halfline a c= dom f & right_closed_halfline a c= dom g
  & f is_+infty_improper_integrable_on a
  & g is_+infty_improper_integrable_on a
  & not (improper_integral_+infty(f,a) = +infty
       & improper_integral_+infty(g,a) = +infty)
  & not (improper_integral_+infty(f,a) = -infty
       & improper_integral_+infty(g,a) = -infty)
holds f-g is_+infty_improper_integrable_on a
 & improper_integral_+infty(f-g,a)
  = improper_integral_+infty(f,a) - improper_integral_+infty(g,a)
proof
    let f,g be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  right_closed_halfline a c= dom f and
A2:  right_closed_halfline a c= dom g and
A3:  f is_+infty_improper_integrable_on a and
A4:  g is_+infty_improper_integrable_on a and
A5:  not (improper_integral_+infty(f,a) = +infty
       & improper_integral_+infty(g,a) = +infty) and
A6:  not (improper_integral_+infty(f,a) = -infty
       & improper_integral_+infty(g,a) = -infty);
A7: dom(-g) = dom g by VALUED_1:8;
A8: -g is_+infty_improper_integrable_on a by A2,A4,Th44;
A9:now assume A10: improper_integral_+infty(f,a) = +infty &
     improper_integral_+infty(-g,a) = -infty; then
     -improper_integral_+infty(g,a) = -(+infty)
       by A2,A4,Th44,XXREAL_3:6;
     hence contradiction by A5,A10,XXREAL_3:10;
    end;
A11:now assume A12: improper_integral_+infty(f,a) = -infty &
     improper_integral_+infty(-g,a) = +infty; then
     -improper_integral_+infty(g,a) = -(-infty)
       by A2,A4,Th44,XXREAL_3:5;
     hence contradiction by A6,A12,XXREAL_3:10;
    end;
    hence f-g is_+infty_improper_integrable_on a
      by A1,A2,A7,A3,A8,A9,Th46;
    improper_integral_+infty(f-g,a)
     = improper_integral_+infty(f,a) + improper_integral_+infty(-g,a)
       by A1,A2,A7,A3,A8,A9,A11,Th46
    .= improper_integral_+infty(f,a) + (-improper_integral_+infty(g,a))
       by A2,A4,Th44;
    hence improper_integral_+infty(f-g,a)
      = improper_integral_+infty(f,a) - improper_integral_+infty(g,a)
        by XXREAL_3:def 4;
end;
