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,a,d) = integral(f,a,c) + 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'];
   per cases;
   suppose A2: not c <= d;
    ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
    c <= b & a <= d by A1,XXREAL_1:1; then
A3: ['d,c'] c= dom f by A1,A2,INTEGR19:2;
    integral(f,a,c) = integral(f,a,d) + integral(f,d,c) by A1,A2,Th1931; then
    integral(f,a,c) - integral(f,d,c)
      = integral(f,a,d) + (integral(f,d,c) - integral(f,d,c)) by RLVECT_1:28
     .= integral(f,a,d) + 0.Y by RLVECT_1:15;
    hence thesis by A3,A2,Th1947;
   end;
   suppose c <= d;
    hence thesis by A1,Th1931;
   end;
end;
