let a, c, b be Real; :: thesis: for f being Function of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['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)) )
let f be Function of REAL,REAL; :: thesis: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] implies ( f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] & integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,b)) ) )
assume A1: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] ) ; :: thesis: ( f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] & integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,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; :: thesis: verum