let a, b, c, d be Real; :: thesis: for Y being RealBanachSpace

for f being continuous PartFunc of REAL, the carrier of Y 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 Y be RealBanachSpace; :: thesis: for f being continuous PartFunc of REAL, the carrier of Y 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 continuous PartFunc of REAL, the carrier of Y; :: 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 ) )

assume A1: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f ) ; :: thesis: ( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded )

- f = (- 1) (#) f by VFUNCT_1:23;

hence - f is_integrable_on ['c,d'] by A1, Th1911; :: thesis: (- f) | ['c,d'] is bounded

f | ['c,d'] is bounded by A1, Th1915b;

hence (- f) | ['c,d'] is bounded by Th1935b; :: thesis: verum

for f being continuous PartFunc of REAL, the carrier of Y 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 Y be RealBanachSpace; :: thesis: for f being continuous PartFunc of REAL, the carrier of Y 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 continuous PartFunc of REAL, the carrier of Y; :: 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 ) )

assume A1: ( a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f ) ; :: thesis: ( - f is_integrable_on ['c,d'] & (- f) | ['c,d'] is bounded )

- f = (- 1) (#) f by VFUNCT_1:23;

hence - f is_integrable_on ['c,d'] by A1, Th1911; :: thesis: (- f) | ['c,d'] is bounded

f | ['c,d'] is bounded by A1, Th1915b;

hence (- f) | ['c,d'] is bounded by Th1935b; :: thesis: verum