reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th47:
  for f be PartFunc of REAL,REAL-NS n st a <= b & ['a,b'] c= dom f
  holds integral(f,b,a) = - integral(f,a,b)
  proof
    let f be PartFunc of REAL,REAL-NS n;
    assume A1: a<=b & ['a,b'] c= dom f;
    then
A2: ['a,b'] = [.a,b.] by INTEGRA5:def 3;
    integral(f,['a,b'])= integral(f,a,b) by A2,INTEGR18:16;
    hence thesis by A2,A1,INTEGR18:18;
  end;
