let n be Element of NAT ; :: thesis: for a, b, c, d being Real
for f being PartFunc of REAL,(REAL n) 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),c,d) = - (integral (f,c,d))
let a, b, c, d be Real; :: thesis: for f being PartFunc of REAL,(REAL n) 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),c,d) = - (integral (f,c,d))
let f be PartFunc of REAL,(REAL n); :: 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),c,d) = - (integral (f,c,d)) )
- f = (- 1) (#) f by NFCONT_4:7;
hence ( 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),c,d) = - (integral (f,c,d)) ) by Th25; :: thesis: verum