
theorem
for f,g be PartFunc of REAL,REAL, a,b be Real st
].a,b.[ c= dom f &
 ].a,b.[ c= dom g & f is_improper_integrable_on a,b &
 g is_improper_integrable_on a,b &
 not (improper_integral(f,a,b) = +infty & improper_integral(g,a,b) = +infty) &
 not (improper_integral(f,a,b) = -infty & improper_integral(g,a,b) = -infty)
 holds
  f-g is_improper_integrable_on a,b &
  improper_integral(f-g,a,b)
   = improper_integral(f,a,b)-improper_integral(g,a,b)
proof
    let f,g be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1: ].a,b.[ c= dom f and
A2: ].a,b.[ c= dom g and
A3: f is_improper_integrable_on a,b and
A4: g is_improper_integrable_on a,b and
A5: not (improper_integral(f,a,b) = +infty
        & improper_integral(g,a,b) = +infty) and
A6: not (improper_integral(f,a,b) = -infty
        & improper_integral(g,a,b) = -infty);
A7: dom(-g) = dom g by VALUED_1:8;
A8: -g is_improper_integrable_on a,b by A2,A4,Th62;
A9:improper_integral(-g,a,b) = - improper_integral(g,a,b) by A2,A4,Th62; then

A10: improper_integral(f,a,b) <> +infty or improper_integral(-g,a,b) <> -infty
      by A5,XXREAL_3:23;

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

    hence f-g is_improper_integrable_on a,b by A1,A2,A3,A7,A8,A10,Th63;
    improper_integral(f-g,a,b)
     = improper_integral(f,a,b) + improper_integral(-g,a,b)
      by A1,A2,A3,A7,A8,A10,A11,Th63
    .=improper_integral(f,a,b) + -improper_integral(g,a,b) by A2,A4,Th62;
    hence improper_integral(f-g,a,b)
      = improper_integral(f,a,b)-improper_integral(g,a,b) by XXREAL_3:def 4;
end;
