reserve X for RealNormSpace;

theorem Th16:
  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 = [.a,b.] 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 A = [.a,b.]; then
A3: a1 = a & b1 = b by A2,INTEGRA1:5; then
  integral(f,a,b) = integral(f,[' a,b ']) by A1,Def9;
  hence thesis by A1,A2,A3,INTEGRA5:def 3;
end;
