
theorem
for f,g be PartFunc of REAL,REAL st dom f = REAL &
 dom g = REAL & f is_improper_integrable_on_REAL &
 g is_improper_integrable_on_REAL &
 not (improper_integral_on_REAL f = +infty
    & improper_integral_on_REAL g = +infty) &
 not (improper_integral_on_REAL f = -infty
    & improper_integral_on_REAL g = -infty)
 holds
  f-g is_improper_integrable_on_REAL &
  improper_integral_on_REAL (f-g)
   = improper_integral_on_REAL f - improper_integral_on_REAL g
proof
    let f,g be PartFunc of REAL,REAL;
    assume that
A1: dom f = REAL and
A2: dom g = REAL and
A3: f is_improper_integrable_on_REAL and
A4: g is_improper_integrable_on_REAL and
A5: not (improper_integral_on_REAL f = +infty
    & improper_integral_on_REAL g = +infty) and
A6: not (improper_integral_on_REAL f = -infty
    & improper_integral_on_REAL g = -infty);
A7: dom(-g) = dom g by VALUED_1:8;
A8: -g is_improper_integrable_on_REAL by A2,A4,Th50;
A9:improper_integral_on_REAL (-g) = -improper_integral_on_REAL g by A2,A4,Th50;

A10: improper_integral_on_REAL f <> +infty
  or improper_integral_on_REAL(-g) <> -infty
      by A5,A9,XXREAL_3:23;

A11: improper_integral_on_REAL f <> -infty
  or improper_integral_on_REAL(-g) <> +infty
      by A6,A9,XXREAL_3:23;

    hence f-g is_improper_integrable_on_REAL by A1,A2,A3,A7,A8,A10,Th51;
    improper_integral_on_REAL(f-g)
     = improper_integral_on_REAL f + improper_integral_on_REAL (-g)
      by A1,A2,A3,A7,A8,A10,A11,Th51; then
    improper_integral_on_REAL(f-g)
     =improper_integral_on_REAL f + -improper_integral_on_REAL g by A2,A4,Th50;
    hence improper_integral_on_REAL (f-g)
      = improper_integral_on_REAL f-improper_integral_on_REAL g
        by XXREAL_3:def 4;
end;
