reserve X for RealNormSpace;

theorem
  for f being PartFunc of REAL,the carrier of X,
      A being non empty closed_interval Subset of REAL,
      a, b be Real
        st A = [.b,a.] & A c= dom f holds -integral(f,A) = integral(f,a,b)
proof
  let f be PartFunc of REAL,the carrier of X;
  let A be non empty closed_interval Subset of REAL;
  let a, b be Real;
  consider a1, b1 being Real such that
A1: a1 <= b1 and
A2: A = [.a1,b1.] by MEASURE5:14;
  assume
A3: A = [.b,a.] & A c= dom f; then
A4: a1 = b & b1 = a by A2,INTEGRA1:5;
  now
    per cases by A1,A4,XXREAL_0:1;
    suppose
A5:   b < a; then
      integral(f,a,b) = -integral(f,[' b,a ']) by Def9;
      hence thesis by A3,A5,INTEGRA5:def 3;
    end;
    suppose
A6:   b = a;
      A = [. lower_bound A,upper_bound A .] by INTEGRA1:4; then
      lower_bound A = b & upper_bound A = a by A3,INTEGRA1:5; then
A7:   vol(A) = upper_bound A - upper_bound A by A6
            .= 0;
A8:   integral(f,a,b) = integral(f,A) by A3,A6,Th16;
A9:   -integral(f,a,b) = -integral(f,A) by A3,A6,Th16;
      integral(f,a,b) = 0.X by A7,A3,Th17,A8;
      hence thesis by A9,RLVECT_1:12;
    end;
  end;
  hence thesis;
end;
