reserve A for non empty closed_interval Subset of REAL;

theorem INTEGR241:
for f being Function of REAL,REAL, a, b, c being Real st
a <= b & b <= c & ['a,c'] c= dom f &
f | ['a,b'] is bounded & f | ['b,c'] is bounded &
f is_integrable_on ['a,b'] & f is_integrable_on ['b,c'] holds
f is_integrable_on ['a,c'] &
integral (f,a,c) = (integral (f,a,b)) + (integral (f,b,c))
proof
 let f be Function of REAL,REAL;
 let a, b, c be Real;
 assume that
 A1: a <= b & b <= c and
 A2: ['a,c'] c= dom f and
 A3: f | ['a,b'] is bounded & f | ['b,c'] is bounded and
 A4: f is_integrable_on ['a,b'] & f is_integrable_on ['b,c'];
 reconsider ff = f as PartFunc of REAL,REAL;
 ['a,c'] c= dom ff &
  ff | ['a,b'] is bounded & ff | ['b,c'] is bounded &
 ff is_integrable_on ['a,b'] & ff is_integrable_on ['b,c'] by A2,A3,A4;
 hence f is_integrable_on ['a,c'] &
 integral (f,a,c) = (integral (f,a,b)) + (integral (f,b,c))
 by INTEGR24:1,A1;
end;
