let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL

for f being Function of A,(REAL n)

for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let A be non empty closed_interval Subset of REAL; :: thesis: for f being Function of A,(REAL n)

for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let f be Function of A,(REAL n); :: thesis: for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let g be Function of A,(REAL-NS n); :: thesis: ( f = g & f is bounded & f is integrable implies ( g is integrable & integral f = integral g ) )

assume A1: ( f = g & f is bounded & f is integrable ) ; :: thesis: ( g is integrable & integral f = integral g )

then A2: g is integrable by Th41;

A3: for T being DivSequence of A

for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds

( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = integral f ) by A1, INTEGR15:11;

reconsider I0 = integral f as Point of (REAL-NS n) by REAL_NS1:def 4;

integral f = I0 ;

then for T being DivSequence of A

for S0 being middle_volume_Sequence of g,T st delta T is convergent & lim (delta T) = 0 holds

( middle_sum (g,S0) is convergent & lim (middle_sum (g,S0)) = I0 ) by A3, A1, Th40;

hence ( g is integrable & integral f = integral g ) by A2, INTEGR18:def 6; :: thesis: verum

for f being Function of A,(REAL n)

for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let A be non empty closed_interval Subset of REAL; :: thesis: for f being Function of A,(REAL n)

for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let f be Function of A,(REAL n); :: thesis: for g being Function of A,(REAL-NS n) st f = g & f is bounded & f is integrable holds

( g is integrable & integral f = integral g )

let g be Function of A,(REAL-NS n); :: thesis: ( f = g & f is bounded & f is integrable implies ( g is integrable & integral f = integral g ) )

assume A1: ( f = g & f is bounded & f is integrable ) ; :: thesis: ( g is integrable & integral f = integral g )

then A2: g is integrable by Th41;

A3: for T being DivSequence of A

for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds

( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = integral f ) by A1, INTEGR15:11;

reconsider I0 = integral f as Point of (REAL-NS n) by REAL_NS1:def 4;

integral f = I0 ;

then for T being DivSequence of A

for S0 being middle_volume_Sequence of g,T st delta T is convergent & lim (delta T) = 0 holds

( middle_sum (g,S0) is convergent & lim (middle_sum (g,S0)) = I0 ) by A3, A1, Th40;

hence ( g is integrable & integral f = integral g ) by A2, INTEGR18:def 6; :: thesis: verum