reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;
reserve X,Y for RealBanachSpace;
reserve E for Point of Y;

theorem Th1947:
  for f be PartFunc of REAL,the carrier of Y st a <= b & ['a,b'] c= dom f
    holds integral(f,b,a) = - integral(f,a,b)
proof
   let f be PartFunc of REAL,the carrier of Y;
   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;
