
theorem
for f,g be PartFunc of REAL,REAL, b be Real
 st left_closed_halfline b c= dom f & left_closed_halfline b c= dom g
  & f is_-infty_improper_integrable_on b
  & g is_-infty_improper_integrable_on b
  & not (improper_integral_-infty(f,b) = +infty
       & improper_integral_-infty(g,b) = +infty)
  & not (improper_integral_-infty(f,b) = -infty
       & improper_integral_-infty(g,b) = -infty)
holds f-g is_-infty_improper_integrable_on b
 & improper_integral_-infty(f-g,b)
  = improper_integral_-infty(f,b) - improper_integral_-infty(g,b)
proof
    let f,g be PartFunc of REAL,REAL, b be Real;
    assume that
A1:  left_closed_halfline b c= dom f and
A2:  left_closed_halfline b c= dom g and
A3:  f is_-infty_improper_integrable_on b and
A4:  g is_-infty_improper_integrable_on b and
A5:  not (improper_integral_-infty(f,b) = +infty
       & improper_integral_-infty(g,b) = +infty) and
A6:  not (improper_integral_-infty(f,b) = -infty
       & improper_integral_-infty(g,b) = -infty);
A7: dom(-g) = dom g by VALUED_1:8;
A8: -g is_-infty_improper_integrable_on b by A2,A4,Th43;
A9:now assume A10: improper_integral_-infty(f,b) = +infty &
     improper_integral_-infty(-g,b) = -infty; then
     -improper_integral_-infty(g,b) = -(+infty) by A2,A4,Th43,XXREAL_3:6;
     hence contradiction by A5,A10,XXREAL_3:10;
    end;
A11:now assume A12: improper_integral_-infty(f,b) = -infty &
     improper_integral_-infty(-g,b) = +infty; then
     -improper_integral_-infty(g,b) = -(-infty)
        by A2,A4,Th43,XXREAL_3:5;
     hence contradiction by A6,A12,XXREAL_3:10;
    end;
    hence f-g is_-infty_improper_integrable_on b
      by A1,A2,A7,A3,A8,A9,Th45;
    improper_integral_-infty(f-g,b)
     = improper_integral_-infty(f,b) + improper_integral_-infty(-g,b)
       by A1,A2,A7,A3,A8,A9,A11,Th45
    .= improper_integral_-infty(f,b) + (-improper_integral_-infty(g,b))
       by A2,A4,Th43;
    hence improper_integral_-infty(f-g,b)
      = improper_integral_-infty(f,b) - improper_integral_-infty(g,b)
       by XXREAL_3:def 4;
end;
