reserve X for RealNormSpace;

theorem Th5:
  for X be RealNormSpace,
      A be non empty closed_interval Subset of REAL,
      f, h be Function of A,the carrier of X
        st h = -f & f is integrable holds
        h is integrable & integral(h) = -integral(f)
proof
  let X be RealNormSpace,
      A be non empty closed_interval Subset of REAL,
   f, h be Function of A,the carrier of X;
  assume A1: h = -f & f is integrable; then
A2: h = (-jj)(#)f by VFUNCT_1:23;
  hence  h is integrable by A1,Th4;
  integral(h) = (-jj)*integral(f) by A1,A2,Th4;
  hence integral(h) = -integral(f) by RLVECT_1:16;
end;
