let n be Element of NAT ; :: thesis: for a, b, c, d being Real
for f being PartFunc of REAL,(REAL n) st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f holds
( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded )
let a, b, c, d be Real; :: thesis: for f being PartFunc of REAL,(REAL n) st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f holds
( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded )
let f be PartFunc of REAL,(REAL n); :: thesis: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f implies ( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded ) )
- f = (- 1) (#) f by NFCONT_4:7;
hence ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f implies ( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded ) ) by Th11; :: thesis: verum