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 Th33:
  integral(f,b,a) = - integral(f,a,b)
  proof
    per cases;
    suppose a <= b; then
A1:   ['a,b'] = [.a,b.] by INTEGRA5:def 3;
      integral(f,['a,b']) = integral(f,a,b) by A1,INTEGR15:19;
      hence integral(f,b,a) = - integral(f,a,b) by A1,INTEGR15:20;
    end;
    suppose not a <= b; then
A2:   ['b,a'] = [.b,a.] by INTEGRA5:def 3;
      then - integral(f,['b,a'])= integral(f,a,b) by INTEGR15:20;
      hence thesis by A2,INTEGR15:19;
    end;
  end;
