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,() 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,() 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,() st f = g & f is bounded & f is integrable holds
( g is integrable & integral f = integral g )

let g be Function of A,(); :: 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 () = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = integral f ) by ;
reconsider I0 = integral f as Point of () 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 () = 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 ; :: thesis: verum