reserve X for RealNormSpace;

theorem
  for X be RealNormSpace,
      A be non empty closed_interval Subset of REAL,
      f, g, h be Function of A,the carrier of X
        st h = f - g & f is integrable & g is integrable holds
        h is integrable & integral(h) = integral(f) - integral(g)
proof
  let X be RealNormSpace,
      A be non empty closed_interval Subset of REAL,
      f, g, h be Function of A,the carrier of X;
  assume A1: h = f - g & f is integrable & g is integrable; then
A2: h = f + (-g) by VFUNCT_1:25;
  dom (-g) = dom g by VFUNCT_1:def 5
          .= A by FUNCT_2:def 1; then
  reconsider gg = -g as Function of A,the carrier of X by FUNCT_2:def 1;
A3: gg is integrable by A1,Th5;
  hence h is integrable by A1,A2,Th6;
  integral(h) = integral(f) + integral(gg) by A1,A2,A3,Th6; then
  integral(h) = integral(f) + -integral(g) by A1,Th5;
  hence integral(h) = integral(f) - integral(g) by RLVECT_1:def 11;
end;
