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 <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) )
let Y be RealBanachSpace; :: thesis: for f being continuous PartFunc of REAL, the carrier of Y st a <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) )
let f be continuous PartFunc of REAL, the carrier of Y; :: thesis: ( a <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies ( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) ) )
assume that
A1: ( a <= b & c <= d ) and
A2: ['a,b'] c= dom f and
A4: ( c in ['a,b'] & d in ['a,b'] ) ; :: thesis: ( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) )
A3: ( f is_integrable_on ['a,b'] & ||.f.|| is_integrable_on ['a,b'] & f | ['a,b'] is bounded ) by Th1, INTEGR20:19, Th4, A1, A2;
['a,b'] = [.a,b.] by A1, INTEGRA5:def 3;
then A5: ( a <= c & d <= b ) by A4, XXREAL_1:1;
then A6: f | ['c,d'] is bounded by A1, A2, A3, Th1915b;
A7: ( ['c,d'] c= dom f & f is_integrable_on ['c,d'] ) by A1, A2, A5, Th1909, INTEGR19:2;
A8: ['a,b'] c= dom ||.f.|| by A2, NORMSP_0:def 2;
||.f.|| | ['a,b'] is bounded by Th3, A1, A2;
then ( ['c,d'] c= dom ||.f.|| & ||.f.|| is_integrable_on ['c,d'] ) by A1, A3, A5, A8, INTEGRA6:18;
hence ( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) ) by A1, A7, Th1921, A6, Th1919; :: thesis: verum