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
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b']
  holds integral(-f,c,d) = -integral(f,c,d)
proof
   let f be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'];
A2:-f = (-1)(#)f by VFUNCT_1:23;
   (-1)*integral(f,c,d)=-integral(f,c,d) by RLVECT_1:16;
   hence thesis by A1,A2,Th1925;
end;
