let a, b, c, d be real number ; :: thesis: for f being PartFunc 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'] & d in ['a,b'] holds
integral f,a,d = (integral f,a,c) + (integral f,c,d)
let f be PartFunc 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'] & d in ['a,b'] implies integral f,a,d = (integral f,a,c) + (integral f,c,d) )
assume A1: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: integral f,a,d = (integral f,a,c) + (integral f,c,d)

now

assume A2: not c <= d ; :: thesis: integral f,a,d = (integral f,a,c) + (integral f,c,d)
then integral f,a,c = (integral f,a,d) + (integral f,d,c) by A1, Lm4;
then A3: integral f,a,d = (integral f,a,c) - (integral f,d,c) ;
integral f,c,d = - (integral f,['d,c']) by A2, INTEGRA5:def 5;
hence integral f,a,d = (integral f,a,c) + (integral f,c,d) by A2, A3, INTEGRA5:def 5; :: thesis: verum

end;

hence integral f,a,d = (integral f,a,c) + (integral f,c,d) by A1, Lm4; :: thesis: verum