let a, b, c, d, e 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'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

|.(f . x).| <= e ) holds

|.(integral (f,c,d)).| <= e * |.(d - c).|

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'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

|.(f . x).| <= e ) implies |.(integral (f,c,d)).| <= e * |.(d - c).| )

set A = ['(min (c,d)),(max (c,d))'];

assume that

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'] ) and

A2: for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

|.(f . x).| <= e ; :: thesis: |.(integral (f,c,d)).| <= e * |.(d - c).|

rng (abs f) c= REAL ;

then A3: abs f is Function of (dom (abs f)),REAL by FUNCT_2:2;

['(min (c,d)),(max (c,d))'] c= dom (abs f) by A1, INTEGRA6:21;

then reconsider g = (abs f) | ['(min (c,d)),(max (c,d))'] as Function of ['(min (c,d)),(max (c,d))'],REAL by A3, FUNCT_2:32;

A4: vol ['(min (c,d)),(max (c,d))'] = |.(d - c).| by INTEGRA6:6;

abs f is_integrable_on ['(min (c,d)),(max (c,d))'] by A1, INTEGRA6:21;

then A5: g is integrable ;

consider h being Function of ['(min (c,d)),(max (c,d))'],REAL such that

A6: rng h = {e} and

A7: h | ['(min (c,d)),(max (c,d))'] is bounded by INTEGRA4:5;

( min (c,d) <= c & c <= max (c,d) ) by XXREAL_0:17, XXREAL_0:25;

then min (c,d) <= max (c,d) by XXREAL_0:2;

then A12: integral ((abs f),(min (c,d)),(max (c,d))) = integral ((abs f),['(min (c,d)),(max (c,d))']) by INTEGRA5:def 4;

(abs f) | ['(min (c,d)),(max (c,d))'] is bounded by A1, INTEGRA6:21;

then A13: g | ['(min (c,d)),(max (c,d))'] is bounded ;

( h is integrable & integral h = e * (vol ['(min (c,d)),(max (c,d))']) ) by A6, INTEGRA4:4;

then integral ((abs f),(min (c,d)),(max (c,d))) <= e * |.(d - c).| by A12, A7, A8, A5, A13, A4, INTEGRA2:34;

hence |.(integral (f,c,d)).| <= e * |.(d - c).| by A11, XXREAL_0:2; :: thesis: verum

|.(f . x).| <= e ) holds

|.(integral (f,c,d)).| <= e * |.(d - c).|

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'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

|.(f . x).| <= e ) implies |.(integral (f,c,d)).| <= e * |.(d - c).| )

set A = ['(min (c,d)),(max (c,d))'];

assume that

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'] ) and

A2: for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

|.(f . x).| <= e ; :: thesis: |.(integral (f,c,d)).| <= e * |.(d - c).|

rng (abs f) c= REAL ;

then A3: abs f is Function of (dom (abs f)),REAL by FUNCT_2:2;

['(min (c,d)),(max (c,d))'] c= dom (abs f) by A1, INTEGRA6:21;

then reconsider g = (abs f) | ['(min (c,d)),(max (c,d))'] as Function of ['(min (c,d)),(max (c,d))'],REAL by A3, FUNCT_2:32;

A4: vol ['(min (c,d)),(max (c,d))'] = |.(d - c).| by INTEGRA6:6;

abs f is_integrable_on ['(min (c,d)),(max (c,d))'] by A1, INTEGRA6:21;

then A5: g is integrable ;

consider h being Function of ['(min (c,d)),(max (c,d))'],REAL such that

A6: rng h = {e} and

A7: h | ['(min (c,d)),(max (c,d))'] is bounded by INTEGRA4:5;

A8: now :: thesis: for x being Real st x in ['(min (c,d)),(max (c,d))'] holds

g . x <= h . x

A11:
|.(integral (f,c,d)).| <= integral ((abs f),(min (c,d)),(max (c,d)))
by A1, INTEGRA6:21;g . x <= h . x

let x be Real; :: thesis: ( x in ['(min (c,d)),(max (c,d))'] implies g . x <= h . x )

assume A9: x in ['(min (c,d)),(max (c,d))'] ; :: thesis: g . x <= h . x

then g . x = (abs f) . x by FUNCT_1:49;

then A10: g . x = |.(f . x).| by VALUED_1:18;

h . x in {e} by A6, A9, FUNCT_2:4;

then h . x = e by TARSKI:def 1;

hence g . x <= h . x by A2, A9, A10; :: thesis: verum

end;assume A9: x in ['(min (c,d)),(max (c,d))'] ; :: thesis: g . x <= h . x

then g . x = (abs f) . x by FUNCT_1:49;

then A10: g . x = |.(f . x).| by VALUED_1:18;

h . x in {e} by A6, A9, FUNCT_2:4;

then h . x = e by TARSKI:def 1;

hence g . x <= h . x by A2, A9, A10; :: thesis: verum

( min (c,d) <= c & c <= max (c,d) ) by XXREAL_0:17, XXREAL_0:25;

then min (c,d) <= max (c,d) by XXREAL_0:2;

then A12: integral ((abs f),(min (c,d)),(max (c,d))) = integral ((abs f),['(min (c,d)),(max (c,d))']) by INTEGRA5:def 4;

(abs f) | ['(min (c,d)),(max (c,d))'] is bounded by A1, INTEGRA6:21;

then A13: g | ['(min (c,d)),(max (c,d))'] is bounded ;

( h is integrable & integral h = e * (vol ['(min (c,d)),(max (c,d))']) ) by A6, INTEGRA4:4;

then integral ((abs f),(min (c,d)),(max (c,d))) <= e * |.(d - c).| by A12, A7, A8, A5, A13, A4, INTEGRA2:34;

hence |.(integral (f,c,d)).| <= e * |.(d - c).| by A11, XXREAL_0:2; :: thesis: verum