 reserve A for non empty Subset of REAL;

theorem
  for a,b,c be Real, f be Function of REAL,REAL st
  a < b & b < c & f is_integrable_on ['a,c'] & f | ['a,c'] is bounded
    holds
  f is_integrable_on ['a,b'] & f is_integrable_on ['b,c'] &
  f | ['a,b'] is bounded & ['a,b'] c= dom f &
  integral (f,a,c) = (integral (f,a,b)) + (integral (f,b,c))
proof
 let a,b,c be Real;
 let f be Function of REAL,REAL;
 assume A1: a < b & b < c; then
 AD: a <= c by XXREAL_0:2;
 assume A2: f is_integrable_on ['a,c'] & f | ['a,c'] is bounded;
 reconsider F = f as PartFunc of REAL,REAL;
 A6:REAL = dom f by FUNCT_2:def 1;
 b in [. a,c .] by A1; then
 A9: b in ['a,c'] by A1,INTEGRA5:def 3,XXREAL_0:2;
 F is_integrable_on ['a,c'] & F | ['a,c'] is bounded by A2;
 hence thesis by A6,A9,INTEGRA6:17,INTEGRA6:18,AD,A1;
end;
