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 Th1908:
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= b & ['a,b'] c= dom f & c in ['a,b'] holds
     f is_integrable_on ['a,c'] & f is_integrable_on ['c,b']
   & integral(f,a,b) = integral(f,a,c) + integral(f,c,b)
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'];
   then ['a,b'] = [.a,b.] by INTEGRA5:def 3; then
A3:a <= c & c <= b by A1,XXREAL_1:1;
   hence f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] by A1,Th1909;
   ['c,b'] c= ['a,b'] by A3,INTEGR19:1; then
A5:['c,b'] c= dom f by A1;
   per cases;
   suppose A6: b = c; then
A7: ['c,b']= [.c,b.] by INTEGRA5:def 3; then
A91:integral(f,c,b) = integral(f,['c,b'] ) by INTEGR18:16;
    ['c,b']= [. lower_bound(['c,b']),upper_bound(['c,b']) .] by INTEGRA1:4;
    then lower_bound(['c,b']) = c & upper_bound(['c,b']) = b by A7,INTEGRA1:5;
    then vol(['c,b']) =0 by A6; then
    integral(f,c,b) = 0.Y by A5,A91,INTEGR18:17;
    hence thesis by A6;
   end;
   suppose b <> c;
    hence thesis by A1,Lm62;
   end;
end;
