let a, b, c, d be Real; :: 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 :: thesis: ( not c <= d implies integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d)) )

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 4;
hence integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d)) by A2, A3, INTEGRA5:def 4; :: thesis: verum

end;

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