reserve A for non empty closed_interval Subset of REAL;

theorem INTEGRA617:
for a, b, c being Real, f being Function of REAL,REAL st
a <= c & f is_integrable_on ['a,c'] &
f | ['a,c'] is bounded & ['a,c'] c= dom f & b in ['a,c'] holds
f is_integrable_on ['a,b'] & f is_integrable_on ['b,c'] &
integral (f,a,c) = (integral (f,a,b)) + (integral (f,b,c))
proof
 let a, c, b be Real;
 let f be Function of REAL,REAL;
 assume A1:a <= b & f is_integrable_on ['a,b'] &
 f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'];
 reconsider ff = f as PartFunc of REAL,REAL;
 ff is_integrable_on ['a,b'] &
 ff | ['a,b'] is bounded & ['a,b'] c= dom ff by A1;
 hence
 f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] &
 integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,b))
 by INTEGRA6:17,A1;
end;
